American Mathematical Society · Trudy Moskov. Matem. Obw. Trans. Moscow Math. Soc. Tom 69 (2008)...

Trudy Moskov. Matem. Obw. Trans. Moscow Math. Soc.Tom 69 (2008) 2008, Pages 27–104

S 0077-1554(08)00169-6Article electronically published on November 19, 2008

ARENS–MICHAEL ENVELOPES, HOMOLOGICAL EPIMORPHISMS,AND RELATIVELY QUASI-FREE ALGEBRAS

ALEXEI YUL’EVICH PIRKOVSKII

Abstract. We describe and investigate Arens–Michael envelopes of associative alge-bras and their homological properties. We also introduce and study analytic analogsof some classical ring-theoretic constructs: Ore extensions, Laurent extensions, andtensor algebras. For some finitely generated algebras, we explicitly describe theirArens–Michael envelopes as certain algebras of noncommutative power series, and

we also show that the embeddings of such algebras in their Arens–Michael envelopesare homological epimorphisms (i.e., localizations in the sense of J. Taylor). For thatpurpose we introduce and study the concepts of relative homological epimorphismand relatively quasi-free algebra. The above results hold for multiparameter quan-tum affine spaces and quantum tori, quantum Weyl algebras, algebras of quantum(2 × 2)-matrices, and universal enveloping algebras of some Lie algebras of smalldimensions.

1. Introduction

Noncommutative geometry emerged in the second half of the last century when it wasrealized that several classical results can be interpreted as some kind of “duality” be-tween geometry and algebra. Informally speaking, those results show that quite often allthe information about a geometric “space” (topological space, smooth manifold, affinealgebraic variety, etc.) is in fact contained in a properly chosen algebra of functions(continuous, smooth, polynomial. . . ) defined on that space. More precisely, this meansthat assigning to a space X the corresponding algebra of functions, one obtains an anti-equivalence between a category of “spaces” and a category of commutative algebras. Insome cases, the resulting category of commutative algebras admits an abstract descrip-tion, which a priori has nothing to do with functions on any space. A typical example isthe first Gelfand–Naimark theorem, asserting that the functor sending a locally compacttopological space X to the algebra C0(X) of continuous functions vanishing at infinityis an anti-equivalence between the category of locally compact spaces and the categoryof commutative C∗-algebras. Another classical result of this nature is the categoricalform of Hilbert’s Nullstellensatz asserting that, over an algebraically closed field k, thefunctor sending an affine algebraic variety X to the algebra of regular (polynomial) func-tions on X is an anti-equivalence between the category of affine varieties over k and thecategory of finitely generated commutative k-algebras without nilpotents.

One of the basic principles of noncommutative geometry states that an arbitrary(noncommutative) algebra should be viewed as an “algebra of functions on a noncom-mutative space”. The reason for this is the fact that many important geometric notionsand theorems, when translated into the “dual” algebraic language, are also true for non-commutative algebras. These kinds of results gave rise to such branches of mathematics

2000 Mathematics Subject Classification. Primary 46M18.The author was supported by the RFFI grant No. 05-01-00982 and No. 05-01-00001 and the President

of Russia’s grant MK-2049.2004.1.

c©2008 American Mathematical Society

27

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

28 A. YU. PIRKOVSKII

as algebraic and operator K-theories, the theory of quantum groups, noncommutativedynamics, etc.

In fact, the term “noncommutative geometry” (as well as the term “geometry”) isa name not for one but for several branches of mathematics which share many similarideas but differ in their objects of study. For example, the natural objects of studyin noncommutative measure theory are von Neumann algebras (since commutative vonNeumann algebras are exactly the algebras of essentially bounded measurable functionson measure spaces), whereas in noncommutative topology one studies C∗-algebras (inconcordance with the mentioned Gelfand–Naimark theorem). Similarly, noncommutativeaffine algebraic geometry mainly deals with finitely generated algebras (without topology)and noncommutative projective algebraic geometry studies graded algebras. Finally, innoncommutative differential geometry one uses dense subalgebras of C∗-algebras withsome special “differential” properties (see [4, 33]).

Notice that the above list of geometric theories misses an important discipline whichshould be placed “between” differential and algebraic geometries, namely, complex-analytic geometry. To the best of the author’s knowledge, there is no theory at themoment that could claim the name of “noncommutative complex-analytic geometry”.Perhaps, one of the main reasons for this is that it is not clear what kind of algebrasshould be viewed as noncommutative generalizations of the algebras of holomorphic func-tions on complex varieties. Of the few papers related to these issues we mention J. Tay-lor’s [67, 68], which consider various completions of a free algebra, and D. Luminet’s[37, 38], dealing with analytic algebras with polynomial identities. Some closely relatedproblems are considered in a series of papers by L. L. Vaksman, S. D. Sinel’shchikov, andD. L. Shklyarov (see, in particular, [71, 63, 70] and [35]), dealing with noncommutativegeneralizations of function theory in classical domains and using methods from the theoryof quantum groups. In particular, in [70], the author introduces a Banach algebra whichis a “quantum” analog of the algebra A(B) of continuous functions on the closed unit ballB which are holomorphic in its interior, and computes its Shilov boundary. A somewhatdifferent approach to “noncommutative complex analysis”, closely related to operatortheory in the spirit of Szekefalvi-Nagy and Fojas, can be traced back to W. Arveson’sclassical papers [2, 3]. It is being systematically developed G. Popescu, K. Davidson,and others (see, for example, [11, 12, 50, 13, 51]).

The algebras considered in the present paper may, for several reasons, be viewed asnoncommutative generalizations of the algebras of holomorphic functions on complexaffine algebraic varieties. They are formally defined as the Arens–Michael envelopes offinitely generated associative algebras. By definition, the Arens–Michael envelope ofa C-algebra A is its completion A with respect to the system of all submultiplicativeprenorms. The Arens–Michael envelopes were introduced by J. Taylor [66, 67] in rela-tion to a problem of multi-operator holomorphic functional calculus. The reason whythose algebras may be relevant to noncommutative complex-analytic geometry is thatthe Arens–Michael envelope of the algebra Oalg(V ) of regular (polynomial) functionson an affine algebraic variety V coincides with the algebra O(V ) of holomorphic func-tions on V endowed with the standard compact-open topology (see Example 3.6 below).Therefore, if one adopts the point of view of noncommutative algebraic geometry andconsiders a finitely generated algebra (for example, some algebra of noncommutativepolynomials1) as an algebra of “regular functions on a noncommutative affine variety”,then its Arens–Michael envelope should be considered as the algebra of “holomorphic

1Following [20], we use the term “algebra of noncommutative polynomials” not as a synonym of theterm “free algebra”, but in a different and somewhat informal sense, meaning an algebra obtained fromthe usual algebra of polynomials in several indeterminates by “deforming” its multiplication.


ARENS–MICHAEL ENVELOPES 29

functions” on the same “variety”. This approach is convenient, in particular, because itis functorial: the correspondence A → A is a functor from the category of algebras to thecategory of Arens–Michael algebras, which is left adjoint to the forgetful functor actingin the opposite direction. Moreover, this functor extends to the category of A-modulesand may therefore be considered as a noncommutative affine analog of the analytizationfunctor [62], widely used in algebraic geometry.

The present paper has two goals. The first is to explicitly compute Arens–Michaelenvelopes for some standard algebras of noncommutative polynomials, frequently usedin noncommutative algebraic geometry. Those include, in particular, Yu. I. Manin’squantum plane, the quantum torus, their higher-dimensional generalizations, quantumWeyl algebras, algebras of quantum matrices, universal enveloping algebras, etc. Wewill show that for many of those algebras of noncommutative polynomials their Arens–Michael envelope can be described as an algebra of noncommutative power series whosecoefficients satisfy certain vanishing conditions at infinity. An interesting point here isthat quite often for algebras of noncommutative polynomials which may be isomorphicas vector spaces (but not as algebras), their Arens–Michael envelopes are not isomorphicas topological vector spaces.

The second goal of this paper, speaking informally, stems from the desire to answer thenatural question of what are the common properties for an algebra of “noncommutativepolynomial functions” and the corresponding algebra of “noncommutative holomorphicfunctions”? In other words, what properties of a finitely generated algebra A are inher-ited, in one sense or another, by its Arens–Michael envelope? Of course, there shouldbe more than one approach to this question, depending on which properties of A aresingled out and on what should be understood by the word “inherited”. The approachused in this paper was suggested by J. Taylor [67] as an essential element of his generalview of “noncommutative holomorphic functional calculus”. It is based on the notion ofhomological epimorphism (or, in Taylor’s own terminology, localization) and allows us tocompare homological properties of A and of A, including in particular their Hochschild(co)homology groups. By definition, a homomorphism of topological algebras A → Bis called a homological epimorphism if the induced functor Db(B) → Db(A) betweenthe bounded derived categories of topological modules is fully faithful. Informally thismeans that all the homological relations between topological B-modules are preservedwhen they are considered as A-modules under the change of rings A → B.

The notion of homological epimorphism (localization) was introduced by Taylor in [67]and played a key role in the construction of the holomorphic calculus for a set of com-muting operators in a Banach space. It looks like, in a purely algebraic setting, homo-logical epimorphisms first appeared almost 20 years later in a paper by W. Geigle and H.Lenzing [17], independently of Taylor’s work and in connection with some questions ofrepresentation theory of finite-dimensional algebras. In particular, the term “homologicalepimorphism” was used for the first time in [17].2 Ten years later homological epimor-phisms were independently rediscovered in a preprint [46] by A. Neeman and A. Ranicki,again in a purely algebraic context, and were used to solve some problems of algebraicK-theory. Notice that [46] uses a different terminology: when A → B is a homologicalepimorphism in the sense of [17], the authors of [46] say that “B is stably flat over A”(see also [48, 49]). Finally, at the end of 2004, R. Meyer [44] introduced, also indepen-dently, homological epimorphisms (therein called “isocohomological morphisms”) in thecontext of bornological algebras. The motivation came from the problem of computingHochschild cohomology groups of various bornological algebras of functions of temperedgrowth on groups (with convolution multiplication) and more general crossed products.

2To the best of the author’s knowledge, this term was suggested by W. Crawley-Boevey.



Thus homological epimorphisms were introduced independently in at least four papersin relation with various problems. That, we believe, indicates that this notion is naturaland is worth further study.

A basic example of homological epimorphism, which motivated the second half ofthis paper, is the canonical embedding of the polynomial algebra C[x1, . . . , xn] (viewedas a topological algebra with respect to the strongest locally convex topology) in theFrechet algebra of entire functions O(Cn). The fact that this embedding is indeed a ho-mological epimorphism (and remains such if Cn is replaced by an arbitrary polydomainU ⊂ Cn), was proved by Taylor in [67] and was substantially used in his constructionof the holomorphic calculus of commuting operators. Since the algebra of entire func-tions is precisely the Arens–Michael envelope of the polynomial algebra, the above resultbrought Taylor to the following natural question: for which algebras A is the canonicalhomomorphism A → A a homological epimorphism? In fact, Taylor considered thatquestion as a first step in his general program of constructing analogs of the holomorphiccalculus of noncommuting operators. He found that the answer to that question is inthe positive when A is a free algebra, and is in the negative when A is the universalenveloping algebra of a semisimple finite-dimensional Lie algebra. The case of universalenveloping algebras was also considered by A. A. Dosiev [16, 15] and the author [48, 49].In particular, it was shown in [16] that the embedding of the universal enveloping algebraU(g) in its Arens–Michael envelope is a homological epimorphism, when g is metabelian(i.e., [g, [g, g]] = 0). Later, A. A. Dosiev generalized this result to nilpotent Lie algebrassatisfying a certain condition of “normal growth” [15], and the author generalized it topositively graded Lie algebras [48]. On the other hand, it was shown in [49] that theembedding of U(g) in its Arens–Michael envelope can be a homological epimorphism onlywhen g is solvable.

In this paper we offer a general method allowing, for many finitely generated alge-bras A, to prove that the canonical homomorphism A → A is a homological epimorphism.It is based on the observation that if A is relatively quasi-free over a subalgebra R andsatisfies certain additional finiteness and projectivity conditions (which can easily bechecked in concrete situations), then A → A is a homological epimorphism wheneverR → R is. This allows us to use induction in problems of that sort. More precisely, ifthe given algebra A has a chain of subalgebras

C = R0 ⊂ R1 ⊂ · · · ⊂ Rn = A,

where each algebra Ri is relatively quasi-free over Ri−1 and satisfies the above additionalfiniteness and projectivity conditions, then it follows that A → A is a homological epi-morphism. This method can be applied to many of the above algebras, in particular,the quantum plane, the quantum torus, some of their multi-dimensional generalizations,quantum Weyl algebras, the algebra of quantum (2× 2)-matrices, the free algebras, andalso, with some modifications, universal enveloping algebras of some solvable Lie algebrasof small dimensions.

Notice that the notion of quasi-free algebra, which plays a key role in the describedmethod, was introduced and studied, in the “absolute” case (i.e., when R = C), byJ. Cuntz and D. Quillen. A “relative” version of this notion, which technically differslittle from the absolute case, is introduced and studied here in Section 7.

Now we describe the contents of the paper. In Section 2 we collect basic facts abouttopological algebras and modules, and also describe a relative version of homology theoryof topological algebras (for the “absolute” version, we refer the reader to [25]; for a purelyalgebraic version of the “relative” theory, see [28]). We also prove, for later use, some



general results about acyclic objects. The Arens–Michael envelopes of topological alge-bras and modules are defined in Section 3, where we also establish their basic propertiesand give the first examples. In Section 4 we introduce some constructions of locally con-vex algebras which are “analytic analogs” of the classical constructions from ring theory:Ore extensions, skew Laurent extensions, and tensor algebras of bimodules. In Section 5we establish connections between the algebraic and analytic constructions from the pre-ceding section. We prove that, under some additional assumptions, the Arens–Michaelenvelope of an Ore extension (resp., skew Laurent extension) is an analytic Ore extension(resp., analytic skew Laurent extension), and that the Arens–Michael envelope of the ten-sor algebra of a bimodule M is an analytic tensor algebra of its Arens–Michael envelopeM. As a consequence, we give explicit descriptions of the Arens–Michael envelopes fora number of algebras of noncommutative polynomials: the universal enveloping algebrasof the two-dimensional solvable Lie algebra and of the three-dimensional Heisenberg Liealgebra, multiparameter quantum affine spaces and quantum tori, quantum Weyl alge-bras, and the algebra of quantum (2 × 2)-matrices. In Section 6 we introduce severalversions of the notion of homological epimorphism: three “relative” ones (left, right,and two-sided) and an absolute one (which coincides with Taylor’s absolute localiza-tion from [67]). We give various characterizations of these notions in the spirit of thosefrom [17]. In Section 7 we introduce relatively quasi-free algebras by giving several equiv-alent definitions (of which the most convenient one is probably the one in the language ofnoncommutative differential forms, cf. [7]) and establish some of their general properties.In particular, we study the action of the Arens–Michael functor on noncommutative dif-ferential forms and on relatively quasi-free algebras, and establish connections with thenotion of two-sided relative homological epimorphism. We also compute the bimodulesof relative differential forms for the algebras considered in Sections 4–5, and prove thatthose algebras are relatively quasi-free. Section 8 is of a technical nature and is devotedto various finiteness conditions for topological algebras and modules, which will be usedlater in the paper. The main results of the paper are contained in Section 9. There westate and prove a general theorem on sufficient conditions for the canonical epimorphismfrom A to its Arens–Michael envelope A to be a homological homomorphism. We alsodiscuss its various consequences and provide concrete applications (see above). Finally,Section 10 is a detailed appendix containing basic facts about derived categories whichare necessary for understating the present paper. We decided to include this appendixbecause all standard texts on derived categories deal with those of abelian categories,which is not enough for our purposes. For details, see [32].

2. Preliminaries

All vector spaces and algebras in this paper are over the field of complex numbers.They are automatically assumed to be associative and unital (i.e., with identity), andall algebra homomorphisms will preserve identity elements. We do not require that1 = 0; in other words, the zero algebra is by definition unital. Modules over algebras arealso assumed to be unital (i.e., the identity of the algebra acts on modules by identitytransformations).

For concepts pertaining to the theory of topological vector spaces see, for example, [60].Notice that in this paper, unlike [60], locally convex spaces (LCS) are not assumed to beHausdorff. The completion of an LCS E is defined as the completion of the associatedHausdorff LCS E/0 and is denoted either E or E . The completed projective tensorproduct of locally convex spaces E and F is denoted E ⊗F .

By a topological algebra we understand a topological space A endowed with a structureof an associative algebra such that the multiplication operator A × A → A, (a, b) → ab



is separately continuous. The category of topological algebras and continuous homomor-phisms will be denoted Topalg. When speaking of continuous homomorphisms betweentopological algebras we will often omit the word “continuous”.

If A is locally convex and complete and the multiplication on A is jointly continuous,then it is called a ⊗-algebra. In this case the multiplication operator A × A → A givesrise to a continuous linear operator mA : A ⊗A → A, which is uniquely determined bythe condition a ⊗ b → ab.

A locally convex algebra is called a Frechet algebra if it is metrizable and complete,i.e., if its underlying LCS is a Frechet space. Since any separately continuous bilinearmap of Frechet spaces is automatically jointly continuous (see [60, III.5.1]), any Frechetalgebra is a ⊗-algebra.

A prenorm ‖ · ‖ on an algebra A is said to be submultiplicative, if ‖ab‖ ≤ ‖a‖‖b‖ forall a, b ∈ A. This is equivalent to saying that the “unit ball” U = a ∈ A : ‖a‖ ≤ 1 withrespect to this prenorm is an idempotent set, i.e., U2 ⊂ U . Therefore submultiplicativeprenorms are precisely Minkowski functionals of convex absorbing circled idempotentsets. In particular, it now easily follows that for any two-sided ideal I ⊂ A and anysubmultiplicative prenorm on A, its quotient prenorm on A/I is also submultiplicative. Alocally convex algebra A is said to be locally multiplicatively-convex (or locally m-convex,or multinormed) if its topology can be defined by a system of submultiplicative prenorms(or, equivalently, if in A there is a basis of convex circled idempotent neighborhoods ofzero). If A is also complete, then it is called an Arens–Michael algebra. Any Banachalgebra is an Arens–Michael algebra, and any Arens–Michael algebra is a ⊗-algebra. Itfollows from the above that a quotient algebra of a locally m-convex algebra (resp., thecompletion of a quotient algebra of an Arens–Michael algebra) is locally m-convex (resp.,is an Arens–Michael algebra).

Notice that any nonzero submultiplicative prenorm on A is equivalent to some submul-tiplicative prenorm ‖ ·‖ such that ‖1‖ = 1 (this is proved the same way as for norms; see,for example, [24, 1.2.5]). Therefore when speaking of nonzero submultiplicative prenormswe will always assume that they satisfy the above identity.

To check whether an algebra is locally m-convex, it is often convenient to use thefollowing lemma from [45].

Lemma 2.1. Let A be a locally convex topological algebra whose topology is defined bya system of prenorms ‖ · ‖ν : ν ∈ Λ. Suppose that for any ν ∈ Λ there are µ ∈ Λ andC > 0 such that ‖a1 · · · an‖ν ≤ Cn‖a1‖µ · · · ‖an‖µ for any a1, . . . , an ∈ A. Then A islocally m-convex.

Corollary 2.2. Let A be a locally convex topological algebra whose topology is definedby a system of prenorms ‖ · ‖ν : ν ∈ Λ. Suppose that for any ν ∈ Λ there are µ ∈ Λand C > 0 such that ‖ab‖ν ≤ C‖a‖µ‖b‖ν for any a, b ∈ A. Then A is locally m-convex.

Let A be a topological algebra, and Mn(A) the algebra of (n × n)-matrices withcoefficients in A. Identifying Mn(A) with An2

, we may endow Mn(A) with the directproduct topology, with respect to which, as can easily be checked, Mn(A) is a topologicalalgebra. If A is locally convex, (resp., locally m-convex, is a Frechet algebra, is an Arens–Michael algebra), then so is Mn(A). Indeed, if ‖ · ‖ν : ν ∈ Λ is a system of prenormsdefining the topology on A, then the topology on Mn(A) is determined by the system ofprenorms

‖a‖ν= max1≤j≤n

n∑i=1

‖aij‖ν

(a = (aij) ∈ Mn(A), ν ∈ Λ

).

A trivial calculation (which we omit) shows that if the prenorms ‖ · ‖ν are submultiplica-tive, then so are ‖ · ‖ν .



A left topological module over a topological algebra A is a topological vector space Xendowed with a structure of a left A-module such that the operator A×X → X, (a, x) →a · x is separately continuous. If A is a ⊗-algebra, the underlying space of the module Xis locally convex and complete, and the above operator is jointly continuous, then X iscalled a left A-⊗-module. In this case we have a well-defined continuous linear operatorA ⊗X → X, a ⊗ x → a · x.

If X and Y are left topological A-modules, then a morphism from X to Y is a con-tinuous linear map ϕ : X → Y, such that ϕ(a · x) = a · ϕ(x) for all a ∈ A and x ∈ X.The morphisms from X to Y form a vector space, denoted Ah(X, Y ). When Y = X thatspace will be denoted simply Ah(X).

Similarly, one defines right topological modules, topological bimodules, and their mor-phisms. The space of morphisms from a right topological A-module X to a right topolog-ical A-module Y (resp., from a topological A-bimodule X to a topological A-bimodule Y )is denoted hA(X, Y ) (resp., AhA(X, Y )).

Suppose X and Y are left topological A-modules. We endow the space Y X of all mapsfrom X to Y with the Tikhonov topology and consider Ah(X, Y ) as a subspace of Y X.The topology on Ah(X, Y ) induced from Y X is called the topology of simple convergence.The subsets of the form

(2.1) M(S, U) =ϕ ∈ Ah(X, Y ) : ϕ(S) ⊂ U

,

where U ⊂ Y is a neighborhood of zero and S ⊂ X is a finite set, form a basis ofneighborhoods of zero in that space. It is easy to check that the canonical isomorphismof vector spaces Ah(A, X) ∼= X, ϕ → ϕ(1) is a topological isomorphism. Similarly,using Xn as notation for the direct sum of n copies of a module X (with the standarddirect sum topology), we have a topological isomorphism Ah(An, X) ∼= Xn. WhenX = An this isomorphism can be viewed as an isomorphism of topological algebrasAh(An) ∼= Mn(A). In particular, if A is an Arens–Michael algebra, then so is Ah(An).

Later on we will give a slight generalization of the last observation (see Lemma 8.2).Suppose A is a ⊗-algebra, X a right A-⊗-module, and Y a left A-⊗-module. A

projective A-module tensor product X ⊗A Y is defined as the completion of the quotientspace of X ⊗Y by the closure of the linear hull of all elements of the form x·a⊗y−x⊗a·y(x ∈ X, y ∈ Y, a ∈ A). For any two continuous prenorms on X and, respectively, onY , their projective tensor product is a prenorm on X ⊗Y and, after factorization andextension by continuity, yields a prenorm on X ⊗A Y, which we shall also call a projectivetensor product of the given prenorms. Notice that, as in a purely algebraic situation, theprojective A-module tensor product admits an abstract categorical description (see [25,II.4.1]).

Recall that the strongest locally convex topology on a vector space E is defined by thesystem of all prenorms on E. We will denote the space E endowed with this topologyby Es. Clearly, if E is an LCS, then its topology is strongest locally convex if and onlyif any linear map from E to any LCS is continuous. Later in this paper we shall oftenuse some basic properties of the strongest locally convex topology. For ease of reference,we collect those properties in a separate proposition.

Proposition 2.3. (i) Any strongest LCS is complete.(ii) If E is a strongest LCS and F ⊂ E is a vector subspace, then the quotient topology

on E/F is the strongest one.(iii) If E is a strongest LCS and F ⊂ E is a vector subspace, then F is closed and

complemented in E and the topology on F induced from E is the strongest one.(iv) If E, F are strongest LCS of at most countable dimension, then any bilinear map

from E × F to an arbitrary LCS is jointly continuous.



(v) Any algebra A of at most countable dimension is a ⊗-algebra with respect to thestrongest locally convex topology and any A-module of at most countable dimen-sion is an A-⊗-module with respect to the strongest locally convex topology.

(vi) Suppose A is a ⊗-algebra, X is a right A-⊗-module, Y is a left A-⊗-module.Suppose also that X and Y are of at most countable dimension and the topologyon each one of them is strongest locally convex. Then the algebraic tensor productX ⊗A Y endowed with the strongest locally convex topology coincides with X ⊗A Y.

Proof. (i) Just notice that E is isomorphic to the topological direct sum of one-dimensional spaces and use the fact that completeness is preserved under directsums (see [60, II.6.2]).

(ii) Just notice that the canonical map from E to (E/F )s is continuous and thereforefactors through E/F .

(iii) Just consider an arbitrary (and automatically continuous) projector from E to Fand use (ii).

(iv) See [5], Corollary A.2.8.(v), (vi) These follow from (iv) and (i).

Remark 2.1. Notice that (v) applies, in particular, to any finitely generated algebra Aand any finitely generated A-module.

Let A be an algebra, X an A-bimodule, and α an endomorphism of A. We define anew structure of an A-bimodule on X by setting ax = a ·x and xa = x ·α(a) for a ∈ A,x ∈ X. This bimodule is denoted Xα. In a similar way we define a bimodule αX. It iseasy to see that if A is a topological algebra (resp., ⊗-algebra), α is continuous, and Xis a topological A-bimodule (resp., A-⊗-bimodule), then Xα and αX are also topologicalA-bimodules (resp., A-⊗-bimodules).

Suppose R is an algebra. Recall that an R-algebra is a pair (A, ηA), where A is analgebra and ηA : R → A is a homomorphism. If, in addition, R and A are topologicalalgebras and ηA is continuous, then (A, ηA) is called a topological R-algebra. Finally, ifboth A and R are ⊗-algebras, then we shall say that (A, ηA) is an R-⊗-algebra. Whenspeaking of R-algebras we shall often abuse the language by saying that A is an R-algebra(without explicitly mentioning ηA); it should not lead to confusion. A similar conventionapplies to topological R-algebras and R-⊗-algebras.

A homomorphism ϕ : A → B between two R-algebras is called an R-homomorphismif ϕηA = ηB . If A is an R-algebra and B is an S-algebra, then an R-S-homomorphismfrom A to B is defined as a pair of homomorphisms f : A → B and g : R → S makingthe diagram

(2.2)A

f B

Rg

ηA

S

ηB

commute. If g is fixed, then we shall sometimes say that f : A → B is an R-S-homomor-phism. The category of topological R-algebras and continuous R-homomorphisms willbe denoted R-Topalg.

Every topological R-algebra (resp., R-⊗-algebra) A will be considered as a topologicalR-bimodule (resp., R-⊗-bimodule) with respect to the homomorphism ηA : R → A.We remark that R-homomorphisms are precisely algebra homomorphisms which are R-bimodule homomorphisms (or in this case, equivalently, homomorphisms of left or rightR-modules).



If A is a topological algebra, and X is a topological A-bimodule, then the directproduct A × X becomes a topological algebra with respect to multiplication

(a, x)(b, y) = (ab, a · y + x · b).

Algebra thus obtained is called the semidirect product of A and X or the trivial extensionof A by X. It is easy to see that if A is a ⊗-algebra, and X is an A-⊗-bimodule, thenA × X is also a ⊗-algebra. If A is a topological R-algebra, then A × X also becomes atopological R-algebra with respect to the homomorphism

ηA×X : R → A × X, ηA×X(r) = (ηA(r), 0).

Notice that the projection p1 : A × X → A to the first factor is an R-homomorphism.Suppose A is an algebra and X is an A-bimodule. Recall that a linear map D : A → X

is called a derivation if D(ab) = D(a) · b + a · D(b) for any a, b ∈ A. Henceforth allderivations of topological algebras with values in topological bimodules will be assumedto be continuous. For a topological algebra A and a topological A-bimodule X thevector space of all derivations from A to X is denoted Der (A, X). If A is a topologicalR-algebra, then a derivation D : A → X is called an R-derivation if DηA = 0. It is notdifficult to see that this means precisely that D is a morphism of R-bimodules (or in thiscase, equivalently, of left or right R-modules). The subspace of Der (A, X) consisting ofR-derivations will be denoted Der R(A, X).

There is a close relationship between R-derivations from A to X and R-homomor-phisms from A to A×X. More precisely, as direct calculation shows, a continuous linearmap ϕ : A → A × X of the form ϕ(a) = (a, D(a)) is an R-homomorphism if and onlyif D is an R-derivation. In other words, there is an isomorphism of vector spaces

(2.3) Der R(A, X) ∼=ϕ ∈ Hom R-Topalg(A, A × X) : p1ϕ = 1A

.

Moreover, the isomorphism (2.3) is in fact an isomorphism between functors of X actingfrom the category of topological A-bimodules to the category of vector spaces.

If A is a ⊗-algebra, then the category of left A-⊗-modules (resp., right A-⊗-modules,A-⊗-bimodules) and (continuous) morphisms between them will be denoted A-mod (resp.,mod-A, A-mod-A). These categories are additive and have both kernels and coker-nels (see [25, 0.4.1]). More precisely, for any morphism ϕ : X → Y its kernel is apair (Ker ϕ, kerϕ), where Kerϕ = ϕ−1(0) (with the topology induced from X) andkerϕ : Ker ϕ → X is the canonical embedding. Similarly, the cokernel of ϕ is a pair(Cokerϕ, cokerϕ), where Coker ϕ is the completion of the quotient module Y/ϕ(X), andcokerϕ : Y → Coker ϕ is the quotient map.

Suppose f : A → B is a homomorphism of ⊗-algebras. Recall that any left B-⊗-module X is a left A-⊗-module under the action a ·x = f(a) ·x (a ∈ A, x ∈ X). Clearly,any morphism of B-⊗-modules ϕ : X → Y is a morphism of A-⊗-modules. Thus wehave a functor from B-mod to A-mod, which we will denote f• and call the restrictionof scalars along f . A similar functor from mod-B to mod-A will also be denoted f•.

Suppose now that (A, ηA) is an R-⊗-algebra. Set A = A-mod, R = R-mod and endowR with the split exact structure (see Appendix, Example 10.1). Let : A → R denotethe restriction-of-scalars functor along ηA; clearly, it is additive and preserves kernelsand cokernels. Hence A-mod can be made an exact category as in Example 10.3 (seeAppendix). The exact category AR thus obtained will be denoted (A, R)-mod (cf. [28]).Thus the admissible sequences in (A, R)-mod are those that split in R-mod. In particular,when R = C, we recover the standard definition of an admissible (or C-split) sequenceof A-⊗-modules used in the homological theory of topological algebras (see [25, 66]).



For a ⊗-algebra A let Aop denote the opposite algebra and Ae the enveloping algebraA ⊗Aop. If A is an R-⊗-algebra and B is an S-⊗-algebra, then we set

mod-(A, R) = (Aop, Rop)-mod,

(A, R)-mod-(B, S) = (A ⊗Bop, R ⊗Sop).

The exact category (A, C)-mod will be denoted simply A-mod; this should not lead toconfusion. We adopt a similar convention for the categories mod-A = mod-(A, C) andA-mod-A = (A, C)-mod-(A, C).

Suppose as before that (A, ηA) is an R-⊗-algebra and let : A-mod → R-mod be therestriction-of-scalars functor along ηA. It has a left adjoint functor, namely, A ⊗R ( · ).Therefore, taking account of Example 10.8 (see Appendix), for any E ∈ R-mod theA-⊗-module A ⊗R E is projective in (A, R)-mod, (A, R)-mod has enough projectives,and a module X is projective in (A, R)-mod if and only if the multiplication morphismA ⊗R X → X is a retraction of left A-⊗-modules.

Modules of the form A ⊗R E, where E ∈ R-mod, will be called free in (A, R)-mod. Forfuture reference we remark that, under the standard identification of right A-⊗-moduleswith left Aop -⊗-modules and of A-⊗-bimodules with left Ae-⊗-modules, free objects inthe exact categories of right A-⊗-modules and A-⊗-bimodules are of the form:

E ⊗R A (E ∈ mod-R) — in mod-(A, R),

A ⊗R E ⊗R A (E ∈ R-mod-R) — in (A, R)-mod-(A, R),

M ⊗R A (M ∈ A-mod-R) — in (A, A)-mod-(A, R),

A ⊗R M (M ∈ R-mod-A) — in (A, R)-mod-(A, A).

Remark 2.2. Of course, all this has an obvious purely algebraic analog (see [28]). Inparticular, if A is an R-algebra, then the exact category of left A-modules (A, R)-algmodis the category of all left A-modules whose admissible sequences are precisely those thatsplit as sequences of left R-modules. Notice that if A and the left A-module X are ofat most countable dimension, then X is projective in (A, R)-algmod if and only if thestrongest locally convex As-⊗-module Xs is projective in (As, Rs)-mod. This followsimmediately from the description of projective objects in terms of free ones (see above)and from the isomorphism (A⊗R X)s

∼= As ⊗RsXs (see Prop. 2.3). Clearly, similar

assertions are also true for right modules and for bimodules.

Since the above categories have enough projective objects, it follows that any addi-tive covariant (resp., contravariant) functor on those categories with values in an exactcategory has a left (resp., right) derived functor (see Appendix).

In particular, let Vect be the abelian category of vector spaces and LCS the quasia-belian category of locally convex spaces (see Appendix, Example 10.2). Then, for a fixedY ∈ (A, R)-mod, the morphism functor

Ah( · , Y ) : (A, R)-mod → Vect

has a right derived functor

RAh( · , Y ) : D−((A, R)-mod) → D+(Vect).

The corresponding n-th classical derived functor is denoted Ext nA,R( · , Y ). Similarly, for

a fixed X ∈ mod-(A, R), the tensor product functor

X ⊗A ( · ) : (A, R)-mod → LCS

has a left derived functor

X ⊗LA( · ) : D−((A, R)-mod) → D−(LCS).



The corresponding n-th classical derived functor is denoted Tor A,Rn (X, · ). When R = C

the constructed functors are denoted Ext nA( · , Y ) and Tor A

n (X, · ), respectively; see [25,III.4].

Remark 2.3. Since the forgetful functor from Vect to the category Ab of abelian groupspreserves kernels and cokernels, it is easy to see that the underlying abelian group ofExt n

A,R(X, Y ) is naturally isomorphic to the group Ext n(A,R)-mod(X, Y ) (see Appendix,

Example 10.9).

For X ∈ mod-(A, R) and Y ∈ (A, R)-mod the canonical morphism X ⊗L

A Y → X ⊗A Y

(see Appendix) gives rise, when passing to cohomology, to a morphism Tor A,R0 (X, Y ) →

X ⊗A Y of LCS. It is an open map onto its image, which is dense in X ⊗A Y, and itskernel is the closure of zero in Tor A,R

0 (X, Y ). We omit the proof because it is identicalto that in the case R = C; see [25, III.4.26]. Therefore, X ⊗A Y can be identified withthe completion of Tor A,R

0 (X, Y ).If B is an S-⊗-algebra and X ∈ (B, S)-mod-(A, R), then X ⊗A ( · ) can be viewed as

a functor from (A, R)-mod to (B, S)-mod. The corresponding left derived functor withvalues in D−((B, S)-mod) will be denoted by the same symbol X ⊗L

A( · ) as before. It willnot lead to confusion because we will always indicate the domain and codomain of thegiven functor.

Remark 2.4. As in a purely algebraic situation (see, for example, [23, I.6] or [30, 1.10]),one can show that RAh and ⊗L

A extend to bifunctors

RAh : D−((A, R)-mod) × D+((A, R)-mod) → D+(Vect),

⊗L

A : D−(mod-(A, R)) × D−((A, R)-mod) → D−(LCS).

Remark 2.5. On the morphism spaces Ah(X, Y ) one can consider various topologies ofuniform convergence on families of bounded subsets (for example, all bounded subsets,or precompact subsets, or finite subsets . . . ); see [60, III.3]. This allows us to view RAhas a functor with values in D+(LCS) and Ext n

A,R( · , · ) as a functor with values in LCS.We will not go into detail since for our purpose, we do not need such a level of generality.

The relative homological dimension of an A-⊗-module X is by definition

dhA,R X = minn ∈ Z+ : Ext n+1

A,R(X, Y ) = 0 ∀Y ∈ (A, R)-mod.

If such an n does not exist, then one sets dhA,R X = +∞. Notice that X is projec-tive in (A, R)-mod if and only if dhA,R X = 0. The relative homological dimensionmay also be defined as the length of a shortest projective resolution (see Appendix)of X in (A, R)-mod. Finally, for any p > dhA,R X and any Y ∈ (A, R)-mod we haveExt p

A,R(X, Y ) = 0. We omit the proofs of these assertions because they are identical tothose in the case R = C (see [25, III.5]).

The relative left global dimension of an R-⊗-algebra A is defined as

dgR A = supdhA,R X : X ∈ (A, R)-mod

.

When R = C the dimensions of modules and algebras just defined are denoted dhA Xand dg A, respectively; see [25, III.5].

We end this section with several results about acyclic objects (see Appendix) whichwill be needed later.

Proposition 2.4. Let A, B and C be exact categories and F : A → B and G : B → C

additive functors having left derived functors. Suppose that the conditions of the theorem



on the composition of derived functors (see Appendix) hold for F and G. Let X ∈ Ob (A)be an F -acyclic object. Then X is GF -acyclic if and only if F (X) is G-acyclic.

Proof. By the assumption, the canonical morphism LF (X) → F (X) is an isomorphismin D−(B). Therefore, (LG LF )(X) → LG(F (X)) is an isomorphism in D−(C). In thecommutative diagram

(2.4)

(LG LF )(X) ∼

LG(F (X))

L(G F )(X) G(F (X))

the top horizontal and the left vertical arrows are isomorphisms. Therefore, the lowerhorizontal arrow is an isomorphism if and only if the right vertical arrow is an isomor-phism. The rest is clear.

Corollary 2.5. Let

AF1 B

R

G1

F2

S

G2

be a commutative diagram of exact categories and additive functors having left derivedfunctors. Suppose that the theorem on the composition of derived functors is true for(F2, G2) and (G1, F1). Suppose X ∈ Ob (R) is acyclic with respect to F2 and G1, andF2(X) is acyclic with respect to G2. Then G1(X) is acyclic with respect to F1.

Proposition 2.6. Any R-⊗-algebra A is acyclic with respect to the tautological functors

(A, R)-mod-(A, R) → (A, A)-mod-(A, R),

(A, R)-mod-(A, R) → (A, R)-mod-(A, A).

Moreover, any projective resolution of A in (A, R)-mod-(A, R) is automatically a projec-tive resolution in (A, A)-mod-(A, R) and (A, R)-mod-(A, A).

Proof. Clearly, it suffices to consider the first of the two functors. Fix a projectiveresolution

(2.5) 0 ← A ← P•

in (A, R)-mod-(A, R). The augmented complex (2.5) in the exact category (A, R)-mod-(R, R) is admissible and all of its objects are projective Therefore, it splits in A-mod-R;i.e., it is admissible in (A, A)-mod-(A, R). This proves the acyclicity. The second assertionfollows at once from the fact that the above functors send free modules to free modulesand therefore projectives to projectives.

Proposition 2.7. Let A be an R-⊗-algebra, B an exact category, and F : A-mod-A → B

an additive covariant functor. Suppose functors

F1 : (A, A)-mod-(A, R) → B,

F2 : (A, R)-mod-(A, R) → B,

F3 : (A, R)-mod-(A, A) → B

act the same way as F . Then LF1(A) ∼= LF2(A) ∼= LF3(A) in D−(B).



Proof. Let I : (A, R)-mod-(A, R) → (A, A)-mod-(A, R) denote the tautological functor.Clearly, F2 = F1 I. In D−(B) we have a chain of morphisms

LF2(A) = L(F1 I)(A) ← LF1(LI(A)) → LF1(I(A)) = LF1(A).

Since I sends projectives to projectives (see the proof of Proposition 2.6), the first arrowin this chain is an isomorphism. By the same proposition, the second arrow is also anisomorphism. Therefore, LF1(A) ∼= LF2(A), as required. The isomorphism LF3(A) ∼=LF2(A) is established similarly.

Corollary 2.8. Let A be an R-⊗-algebra and X an A-⊗-bimodule. Then for any n ∈ Z,we have isomorphisms of vector spaces

Tor Ae, Re

n (X, A) ∼= Tor Ae, A ⊗Rop

n (X, A) ∼= Tor Ae, R ⊗Aop

n (X, A),

Ext nAe, Re(A, X) ∼= Ext n

Ae, A ⊗Rop(A, X) ∼= Ext nAe, R ⊗Aop(A, X).

Remark 2.6. If we endow the Ext -spaces from Corollary 2.8 with the locally convextopology as in Remark 2.5, then the above isomorphisms will also be topological isomor-phisms.

Definition 2.1. Suppose A is an R-⊗-algebra and X is an A-⊗-bimodule. The vectorspaces

Hn(A, R; X) = Tor Ae, Re

n (X, A), Hn(A, R; X) = Ext nAe, Re(A, X)

are called, respectively, the n-th relative Hochschild homology (cohomology) group of Awith coefficients in X.

Definition 2.2. The number dbR A = dhAe, Re A is called the relative bidimension ofthe R-⊗-algebra A is. When R = C, it is denoted dbA (see [25, III.5]).

Remark 2.7. In view of Corollary 2.8, the above definitions of relative homology andcohomology and relative bidimension are equivalent to Hochschild’s original definitionsin [28]. See also [19].

3. Arens–Michael envelopes

Arens–Michael envelopes (under a different name) were introduced in [66]. Here weadopt the terminology from [24].

Definition 3.1. Suppose A is a topological algebra. A pair (A, ιA) consisting of anArens–Michael algebra A and a continuous homomorphism ιA : A → A is called theArens–Michael envelope of A if for any Arens–Michael algebra B and any continuoushomomorphism ϕ : A → B there exists a unique continuous homomorphism ϕ : A → Bmaking the following diagram commute:

(3.1)A

ϕ B

A

ιA

ϕ

We then say that ϕ extends ϕ (even though ιA is in general not injective; see Example 3.4below).

In other words, if AM denotes the full subcategory of Topalg consisting of Arens–Michael algebras and Set denotes the category of sets, then the Arens–Michael envelopeof a topological algebra A is a representing object for the functor Hom Topalg(A, · ) : AM →Set. The Arens–Michael envelope is clearly unique in the sense that if (A, ιA) and (A, jA)



are Arens–Michael envelopes of the same topological algebra A, then there exists a uniqueisomorphism of topological algebras ϕ : A → A making the following diagram commute:

Aϕ A

A

ιA

jA

Remark 3.1. Notice that it suffices to check the universal property of the Arens–Michaelenvelope expressed by diagram (3.1) only for homomorphisms with values in Banachalgebras. This follows easily from the fact that each Arens–Michael algebra is the inverselimit of a system of Banach algebras (see, for example, [24, Ch. V]). Indeed, suppose A isa topological algebra, A is an Arens–Michael algebra, and ιA : A → A is a homomorphismsuch that for any Banach algebra B and any homomorphism ϕ : A → B there is a uniquehomomorphism ϕ : A → B making diagram (3.1) commutative. Take an arbitrary Arens–Michael algebra C and represent it as the inverse limit lim←−Cν of a system of Banachalgebras. Then

Hom Topalg(A, C) = Hom Topalg(A, lim←−Cν) = lim←−Hom Topalg(A, Cν)

= lim←−Hom AM(A, Cν) = Hom AM(A, lim←−Cν) = Hom AM(A, C).

Therefore, (A, ιA) is the Arens–Michael envelope of A.

Every topological algebra A has an Arens–Michael envelope (see [66] or [24, Ch. V]).More precisely, (A, ιA) can be obtained as the completion of A with respect to a newtopology defined by the family of all continuous submultiplicative prenorms on A. Inparticular, this implies that the image of the canonical homomorphism ιA : A → A isdense. It is easy to see that the correspondence A → A is a functor from Topalg to AMand that it is the left adjoint of the embedding functor from AM to Topalg. Henceforthwe shall call it the Arens–Michael functor.

If A is an algebra without topology, then we define its Arens–Michael envelope as theone with respect to the strongest locally convex algebra As (see Section 2).

We now give several basic examples of Arens–Michael envelopes. Other examples willbe given in Section 5.

Example 3.1. Suppose the polynomial algebra A = C[z1, . . . , zn], is endowed with thestrongest locally convex topology. Taylor [66] noticed that its Arens–Michael envelopecoincides with the algebra of entire functions O(Cn) endowed with the compact-opentopology and that the homomorphism ιA is the tautological embedding. Since this ex-ample serves as the motivation for the rest of the paper, we sketch the basic idea behindthe proof of the last assertion. For simplicity, we consider the case n = 1.

The Cauchy inequality for the coefficients of a power series implies that the compact-open topology on O(C) can be defined by the family of norms

‖f‖ρ =∑

i

|ci|ρi(f =∑

i

cizi ∈ O(C); 0 < ρ < ∞

).

For any Banach algebra B and homomorphism ϕ : C[z] → B, set b = ϕ(z). Then for anyentire function f =

∑i ciz

i ∈ O(C) the series∑

i cibi converges absolutely in B to some

f(b) ∈ B; moreover, ‖f(b)‖ ≤ ‖f‖ρ for any ρ ≥ ‖b‖. The map f → f(b) is the desiredhomomorphism from O(C) to B extending ϕ.

The case of an arbitrary n is argued similarly.



Example 3.2. Let A = Fn be the free C-algebra with n generators ζ1, . . . , ζn. For anyset of integers α = (α1, . . . , αk) from [1, n], let ζα be the element ζα1 · · · ζαk

of Fn. Itis convenient to assume that the identity of the algebra corresponds to the set of zerolength (k = 0). We also set |α| = k. The algebra Fn of noncommutative power seriesconsists of all formal expressions a =

∑α λαζα (where λα ∈ C) such that

‖a‖ρ =∑α

|λα|ρ|α| < ∞ for all 0 < ρ < ∞.

The system of prenorms ‖ · ‖ρ : 0 < ρ < ∞ makes Fn into a Frechet–Arens–Michaelalgebra. Clearly, Fn contains Fn as a subalgebra. Taylor showed [67] (cf. also theprevious example), that Fn is the Arens–Michael envelope of Fn. Notice that whenn = 1, we have F1 = C[z] and F1

∼= O(C).

Example 3.3. Let g be a semisimple Lie algebra and A = U(g) its universal envelopingalgebra. Taylor showed [67] that the Arens–Michael envelope of U(g) is topologicallyisomorphic to the direct product of a countable family of full matrix algebras. More pre-cisely, each finite-dimensional irreducible representation πλ of g yields a homomorphismϕλ : U(g) → Mdλ

(C) (here dλ is the dimension of πλ). Let g be the set of equivalenceclasses of irreducible finite-dimensional representations of g; we then have a homomor-phism ∏

λ∈g

ϕλ : U(g) →∏λ∈g

Mdλ(C).

The algebra∏

λ Mdλ(C) endowed with the direct product topology together with the

above homomorphism is the Arens–Michael envelope of U(g).

Notice that for the algebras in Examples 3.1–3.3 the canonical homomorphism ιA tothe Arens–Michael envelope is injective. The reason for this is that those algebras havesufficiently many Banach representations. On the other hand, it may happen that a givenalgebra A has no nonzero Banach representations, and in that case its Arens–Michaelenvelope is trivial. For illustration, consider the following example.

Example 3.4. Suppose A is a Weyl algebra (endowed, as the algebras from the aboveexamples, with the strongest locally convex topology). Recall that it is generated (asa unital C-algebra) by generators ∂ and x, subject to the relation [∂, x] = 1. It iswell known (see, for example, [59, 13.6]) that a nonzero normed algebra cannot containelements ∂ and x with such a property. Hence there are no nonzero submultiplicativeprenorms on A and therefore A = 0.

Other examples of algebras with trivial Arens–Michael envelopes can be found in [24,5.2.26].

Example 3.5. Choose any Banach algebra A such that all homomorphisms from it toany other Banach algebra are continuous. For example, this could be the algebra B(H)of bounded operators in a Hilbert space or, more generally, in a Banach space E suchthat E ∼= E⊕E; see [10, 5.4.12]. Endowing A with the strongest locally convex topologywe have the topological algebra As. Then, clearly, we have an isomorphism As

∼= A.

When describing Arens–Michael envelopes it is often convenient to use the fact that theArens–Michael functor commutes with the quotient operation on algebras. The followingproposition is proved in [48].

Proposition 3.1. Suppose A is a topological algebra and I is a two-sided ideal in A.Let J denote the closure of the set ιA(I) in A. Then J is a two-sided ideal in A, and



the homomorphism A/I → A/J induced by the homomorphism ιA : A → A extends toan isomorphism of topological algebras

A/I ∼= (A/J)∼.

Since the quotient space of any Frechet space by a closed subspace is complete, wehave the following.

Corollary 3.2. Under the assumptions of Proposition 3.1, assume in addition that A isa Frechet algebra. Then A/I ∼= A/J .

Corollary 3.3. If A is a finitely generated algebra, then As is a trace class Frechetalgebra.

Proof. Since A is finitely generated, it is isomorphic to a quotient algebra of the freealgebra Fn with finitely many generators. Now apply Example 3.2 and Corollary 3.2, aswell as the fact that the Frechet algebra Fn is trace class (see [37]). Example 3.6. Suppose V is an affine complex algebraic variety embedded in CN . LetOalg(V ) denote the algebra of regular (polynomial) functions on V. Generalizing Exam-ple 3.1, we shall show that its Arens–Michael envelope is the Frechet algebra of holomor-phic functions O(V ). To this end, notice that Oalg(V ) = Oalg(CN )/I, where I is the idealin Oalg(CN ) of all functions vanishing on V. Since the algebra Oalg(CN ) = C[z1, . . . , zN ]is Noetherian, there is an exact sequence

(3.2)(Oalg(CN )

)r → Oalg(CN ) → Oalg(V ) → 0

of Oalg(CN )-modules. It gives rise to an exact sequence

(3.3) (OalgCN )r → O

algCN → i∗O

algV → 0

of sheaves of OalgCN -modules, where i denotes the embedding of V in CN . Applying the

analytization functor to (3.3) (see, for example, [69, 13.4] or [62]), we have an exactsequence

(OCN )r → OCN → i∗OV → 0of OCN -modules. Finally, applying the global section functor we have, by Cartan’s The-orem B, an exact sequence

(3.4)(O(CN )

)r → O(CN ) → O(V ) → 0

of O(CN )-modules. Let J ⊂ O(CN ) denote the ideal of all functions vanishing on V.Comparing (3.2) and (3.4), we see that if I ⊂ Oalg(CN ) is generated by r functions, thenthe same functions generate J as an ideal in O(CN ). Therefore, J is the smallest idealin O(CN ) containing I. In particular, J ⊂ I. Since the inverse inclusion is obvious bythe closedness of J , we have J = I. Now it remains to apply Corollary 3.2, according towhich (

Oalg(V ))

=(Oalg(CN )/I

)= O(CN )/J = O(V ).

Next, we shall establish several auxiliary results about modules over Arens–Michaelalgebras. Suppose A is an algebra and X is a left A-module. We shall say that aprenorm ‖ · ‖X on X is m-compatible if there is a submultiplicative prenorm ‖ · ‖A on Asuch that ‖a · x‖X ≤ ‖a‖A‖x‖X for all a ∈ A and x ∈ X. A similar definition makessense when X is a right module or a bimodule. In the latter case we shall simultaneouslyrequire that ‖a · x‖X ≤ ‖a‖A‖x‖X and ‖x · a‖X ≤ ‖x‖X‖a‖A for all a ∈ A and x ∈ X.If A is a topological algebra and X is a topological A-module, then we shall also requirethat both prenorms ‖ · ‖X and ‖ · ‖A be continuous. Notice that if Y is a submoduleof X, then the quotient prenorm on X/Y induced by an m-compatible prenorm on X isalso m-compatible.



Proposition 3.4. Suppose A is an Arens–Michael algebra and X is a left A-⊗-module.Then the topology on X can be defined by some family of m-compatible prenorms.

Proof. Since every left A-⊗-module is a quotient of a free module, it suffices to provethe assertion for free modules. Thus let X = A ⊗E be a free A-module. For eachcontinuous submultiplicative prenorm on A and each continuous prenorm on E theirprojective tensor product is clearly an m-compatible prenorm on X. On the other hand,the prenorms of this form do define the topology on X. Remark 3.2. Suppose ‖·‖λ : λ ∈ Λ1 is a directed system of submultiplicative prenormsdefining the topology on A and ‖·‖µ : µ ∈ Λ2 is a directed system of prenorms definingthe topology on X. For each λ ∈ Λ1 and each µ ∈ Λ2 we define a new prenorm on X bysetting

‖x‖λ,µ = inf n∑

i=1

‖ai‖λ‖xi‖µ : x =n∑

i=1

aixi, ai ∈ A, xi ∈ X, n ∈ N

.

It is easy to check that this prenorm is m-compatible and that the system of all prenormson X of this form is equivalent to the original one.

Remark 3.3. Notice that the completeness of the algebras and modules in Proposition 3.4is not essential.

Proposition 3.5. Suppose that A is an Arens–Michael algebra. Then any left A-⊗-module X is isomorphic to the inverse limit of some system of Banach A-⊗-modules.

Proof. For each m-compatible prenorm ‖ · ‖X on X the set of x ∈ X such that ‖x‖X = 0is a submodule of X. With the aid of Proposition 3.4, the rest of the proof is arguedsimilarly to the proof in the case A = C (see, for example, [60, II.5.4]). Remark 3.4. Assertions similar to Propositions 3.4 and 3.5 for right modules and forbimodules also hold.

Suppose E is a locally convex space whose topology is defined by a system of prenorms‖ · ‖λ : λ ∈ Λ. We can make Λ a partially ordered set by defining λ ≺ µ if ‖x‖λ ≤ ‖x‖µ

for all x ∈ E. Henceforth we shall assume that the indexing set for prenorms on a locallyconvex space is ordered this way.

Proposition 3.6. Suppose A is an Arens–Michael algebra and X is an A-⊗-bimodule.Then the trivial extension A × X is also an Arens–Michael algebra.

Proof. Let ‖ · ‖λ : λ ∈ Λ1 be the system of all continuous submultiplicative prenormson A and ‖ · ‖µ : µ ∈ Λ2 the system of all continuous m-compatible prenorms on X.In view of Proposition 3.4, these systems define the topologies on A and X, respectively.For each λ ∈ Λ1 and each µ ∈ Λ2 we define a prenorm on A×X by setting ‖(a, x)‖λ,µ =‖a‖λ+‖x‖µ. Clearly, the system of all such prenorms defines the direct product topologyon A × X. Using the definition of an m-compatible prenorm, for any µ ∈ Λ2 we choosean element ϕ(µ) ∈ Λ1 such that ‖a · x‖µ ≤ ‖a‖ϕ(µ)‖x‖µ and ‖x · a‖µ ≤ ‖x‖µ‖a‖ϕ(µ) forany a ∈ A and any x ∈ X. Set Λ = (λ, µ) ∈ Λ1 × Λ2 : ϕ(µ) ≺ λ. Clearly, Λ is cofinalin Λ1 × Λ2 and therefore the topology on A × X is defined by the system of prenorms‖ · ‖λ,µ : (λ, µ) ∈ Λ. For any elements (a, x) and (b, y) in A×X and any (λ, µ) ∈ Λ wehave

‖(a, x)(b, y)‖λ,µ = ‖(ab, a · y + x · b)‖λ.µ

= ‖ab‖λ + ‖a · y + x · b‖µ ≤ ‖a‖λ‖b‖λ + ‖a‖λ‖y‖µ + ‖x‖µ‖b‖λ

≤ (‖a‖λ + ‖x‖µ)(‖b‖λ + ‖y‖µ) = ‖(a, x)‖λ,µ‖(b, y)‖λ,µ.



Therefore, each prenorm ‖ · ‖λ,µ for (λ, µ) ∈ Λ is submultiplicative and A × X is anArens–Michael algebra.

Using the preceding result it is not difficult to show that the Arens–Michael envelopehas the extension property not just for homomorphisms but also for derivations definedon the original algebra. Before giving a rigorous statement and a proof of this assertionwe want to make some observations. Suppose A is a topological R-algebra, B is atopological S-algebra, and (f, g) is an R-S-homomorphism from A to B (see (2.2)). It iseasy to see that for any topological B-bimodule X and any S-derivation D : B → X thecomposition D f : A → X is an R-derivation. Thus we have a map

(3.5) Der S(B, X) → Der R(A, X), D → D f,

obviously giving rise to a morphism of functors of X acting from the category of topo-logical B-bimodules to the category of vector spaces.

Recall that a morphism f : X → Y in an arbitrary category A is said to be anepimorphism if for any Z ∈ Ob (A) and any g, h : Y → Z we have g = h whenevergf = hf . If this condition is only satisfied for those Z which are objects of a certain fullsubcategory C ⊂ A, then we shall say that f is an epimorphism relative to C. For easeof reference we state the following lemma, the proof of which is obvious.

Lemma 3.7. Suppose that in some category A we are given a diagram

(3.6)A

f Bh C

Rg

α

S

γ

β

with a commutative left square. Suppose further that the diagram obtained by removingβ is also commutative. If g is an epimorphism relative to some full subcategory C ⊂ A

containing C, then the whole diagram (3.6) is commutative.

Proposition 3.8. Suppose A, R, S are topological algebras and g : R → S is an epi-morphism in Topalg relative to a subcategory of the category of Arens–Michael algebrasAM. Suppose that A is a topological R-algebra and its Arens–Michael envelope A is atopological S-algebra such that the pair (ιA, g) is an R-S-homomorphism from A to A.Then for any A-⊗-bimodule X the canonical map

(3.7) Der S(A, X) → Der R(A, X), D → D ιA

is a bijection. In other words, any R-derivation from A to X uniquely extends to anS-derivation from A to X.

Proof. The injectivity of (3.7) is obvious since Im ιA is dense in A. To establish the sur-jectivity, take an arbitrary R-derivation D : A → X and construct an R-homomorphismϕ : A → A×X, ϕ(a) = (a, D(a)) (see (2.3)). Composing it with the R-S-homomorphismιA × 1X : A × X → A × X we have an R-S-homomorphism ψ : A → A × X. SinceA × X is an Arens–Michael algebra (Proposition 3.6), there exists a unique continuoushomomorphism ψ : A → A × X extending ψ. Thus for any a ∈ A we have

(3.8) ψ(ιA(a)) = ψ(a) = (ιA(a), D(a)).

Since Im ιA is dense in A it follows that ψ(b) = (b, D(b)) for some uniquely determinedmap D : A → X. This map is a derivation by virtue of (2.3) and we also have D ιA = D



because of (3.8). It remains to show that D is an S-derivation, or, equivalently, that ψis an S-homomorphism. To this end, consider the diagram

(3.9)A

ιA Aψ A × X

Rg

ηA

S

ηA×X

ηA

Its left square is commutative by assumption. The diagram obtained from it by removingηA is also commutative, because ψιA = ψ is an R-S-homomorphism. Thus we are underthe assumptions of Lemma 3.7 and therefore the right triangle in (3.9) is commutative,i.e., ψ is an S-homomorphism.

Remark 3.5. Clearly, any homomorphism between topological algebras with a denseimage is an epimorphism relative to AM. Further results about epimorphisms betweentopological algebras will be mentioned in Section 6.

Corollary 3.9. For any topological R-algebra A and any A-⊗-bimodule X the canonicalmap Der R(A, X) → Der R(A, X), D → DιA is a bijection.

Later on we will need a notion of the Arens–Michael envelope for modules. For def-initeness, we only consider left modules; the cases of right modules and bimodules aredealt with similarly.

Definition 3.2. Let A be a topological algebra, A its Arens–Michael envelope, andX a left topological A-module. A pair (X, ιX) consisting of a left A-⊗-module X anda continuous morphism ιX : X → X of A-modules is called the Arens–Michael envelopeof X if for any left A-⊗-module Y and any continuous morphism α : X → Y of A-modules,there is a unique continuous morphism α : X → Y of A-⊗-modules making the followingdiagram commute:

(3.10)X

α Y

X

ιX

α

We shall also say that α extends α.

To construct the Arens–Michael envelope of an A-module X, we define a new topologyon X using the system of all continuous m-compatible prenorms. The LCS thus obtainedwill be denoted Xm and its completion, X. We obviously have a continuous linear mapιX : X → X. Furthermore, let Am denote the algebra A with topology defined by thesystem of all continuous submultiplicative prenorms. Recall that its completion is theArens–Michael envelope A of A. It easily follows from the definition of an m-compatibleprenorm that Xm is a left topological Am-module and that the action of Am on Xm

is jointly continuous. We conclude that there is a unique structure of left A-⊗-moduleon X such that the canonical map Xm → X is a morphism of Am-modules. Therefore,ιX : X → X is a morphism of A-modules.

Proposition 3.10. The pair (X, ιX) constructed above is the Arens–Michael envelopeof X.

Proof. Suppose Y is a left A-⊗-module and α : X → Y is a morphism of A-modules. Thenfor any m-compatible prenorm ‖·‖ on Y the prenorm x → ‖α(x)‖ on X is also, as is easy



to check, m-compatible. Since the topology on Y is defined by m-compatible prenorms(Proposition 3.4), we see that α is continuous as a map from Xm to Y. Therefore, thereis a unique continuous linear map α : X → Y extending α. The fact that α is a morphismof A-modules is obvious since the images of ιA in A and of ιX in X are dense.

Remark 3.6. The Arens–Michael envelope of a left topological A-module can also beobtained using the construction mentioned in the proof of Proposition 3.5.

As in the case of algebras, it is easy to see that the correspondence X → X is a functorfrom the category of left topological A-modules to the category of left A-⊗-modules,which is left adjoint to the restriction-of-scalars functor along the homomorphism ιA. Inother words, for any left topological A-module X and any left A-⊗-module Y , we havea natural (in X and Y ) isomorphism

(3.11) Ah(X, Y ) ∼= Ah(X, Y ).

Notice that the functor X → X is additive.From (3.11) we immediately have the following.

Proposition 3.11. If A is a ⊗-algebra and X a left A-⊗-module, then there is a naturalisomorphism of left A-⊗-modules

(3.12) X ∼= A ⊗A X.

Remark 3.7. It is trivial to check that if A is a topological algebra, then the Arens–Michael envelope (A, ιA) is also the Arens–Michael envelope of A as a left A-module.Therefore, for any n ∈ N , the Arens–Michael envelope of the free A-module An is thepair (An, ιnA).

The next proposition is a “module-theoretic analog” of Proposition 3.1.

Proposition 3.12. Suppose A is a topological algebra, X is a left topological A-module,and Y ⊂ X is a submodule. Let Z denote the closure of ιX(Y ) in X. Then Z is an A-submodule of X and the morphism X/Y → X/Z induced by the morphism ιX : X → X

extends to an isomorphism of A-⊗-modules

X/Y ∼= (X/Z)∼.

In particular, if X is metrizable, then X/Y ∼= X/Z.

Proof. That Z is an A-submodule of X follows from the fact that Im ιX is dense in X.For any left A-⊗-module W we have natural isomorphisms

Ah(X/Y, W ) ∼=ϕ ∈ Ah(X, W ) : ϕ

∣∣Y

= 0

∼=ψ ∈ Ah(X, W ) : ψ

∣∣ιX(Y )

= 0

=ψ ∈ Ah(X, W ) : ψ

∣∣Z

= 0

∼= Ah(X/Z, W ) ∼= Ah((X/Z)∼, W

).

Moreover, for W = (X/Z)∼, the morphism X/Y → W mentioned in the statement ofthe proposition corresponds to the identity morphism 1W . The rest is clear.

Remark 3.8. The definition and construction of the Arens–Michael envelope of left mod-ules carries over, with obvious modifications, to the case of right modules or bimodules.In particular, in the case of bimodules, the natural isomorphism (3.12) becomes

(3.13) X ∼= A ⊗A X ⊗A A.



4. Analytic analogs of classical ring-theoretic constructions

4.1. Analytic Ore extensions. We recall some standard facts about Ore extensions(see, for example, [31, 1.7]). Suppose R is an arbitrary associative algebra (withouttopology) and α : R → R is an endomorphism. A linear map δ : R → R is called an α-derivation if δ(ab) = δ(a)b+α(a)δ(b) for any a, b ∈ R. In other words, δ is an α-derivationif and only if it is a derivation with values in the R-bimodule αR (see Section 2). The Oreextension (or the algebra of skew polynomials) R[t; α, δ] as a left R-module consists of allpolynomials

∑ni=0 rit

i in t with coefficients in R. Multiplication on R[t; α, δ] is uniquelydetermined by the relation tr = α(r)t + δ(r) for r ∈ R and the requirements that thenatural embeddings R → R[t; α, δ] and C[t] → R[t; α, δ] be algebra homomorphisms.Notice that the former embedding makes R[t; α, δ] an R-algebra. The algebra R[t; α, δ]and its element t have the following universal property, which completely characterizesthem: for any R-algebra A and any x ∈ A such that x · r = α(r) · x + ηA(δ(r)) (r ∈ R),there exists a unique R-homomorphism f : R[t; α, δ] → A such that f(t) = x. Whenα = 1R, the algebra R[t; α, δ] is denoted R[t; δ], and when δ = 0 it is denoted R[t; α].

Next, we recall a useful formula for the multiplication on R[t; α, δ]. For each k, n ∈ Z+

and each k ≤ n, let Sn,k : R → R denote the operator which is the sum of all compositionsof k copies of δ and n − k copies of α (with a total of

(nk

)summands). Then for any

r ∈ R we have

(4.1) tnr =n∑

k=0

Sn,k(r)tn−k.

In particular, when δ = 0 the last formula turns into tnr = αn(r)tn, and when α = 1R

it becomes

tnr =n∑

k=0

(n

k

)δk(r)tn−k.

Later on we shall construct a locally convex analog of R[t; α, δ], called an analytic Oreextension O(C, R; α, δ). In Section 5 we shall show that, under some additional assump-tions, the Arens–Michael envelope of an Ore extension is an analytic Ore extension.

Suppose E is a vector space and T is a family of linear operators in E.

Definition 4.1. A prenorm ‖ · ‖ on E will be called T-stable if for any T ∈ T there is aC > 0 such that ‖Tx‖ ≤ C‖x‖ for all x ∈ X.

Clearly, a scalar multiple of a T-stable prenorm and the sum of two T-stable prenormsis T-stable. It is also easy to see that if a subfamily S ⊂ T is such that T is contained inthe subalgebra generated by S, then any S-stable prenorm is also T-stable.

Suppose now that E is a locally convex space.

Definition 4.2. A family T ⊂ L(E) is said to be localizable if the topology on E can bedefined by a family of T-stable prenorms. An operator T ∈ L(E) is said to be localizable3

if T is.

If E is complete, then it is not difficult to see that T is localizable if and only if E isthe inverse limit lim←−Eν of Banach spaces Eν (ν ∈ Λ) such that each operator T ∈ T isof the form T = lim←−Tν for some compatible family of operators Tν ∈ L(Eν) : ν ∈ Λ.This was the motivation for the term “localizable family”.

Notice that the notion of localizable operator is closely related to the notion of tameoperator (see [22]). More precisely, it is easy to see that an operator T in a Frechet

3Operators with this property were independently introduced by W. Zelazko [73] under the name ofm-topologizable operators.



space E is localizable if and only if E admits a grading (in the sense of [22], i.e., anincreasing defining sequence of prenorms) such that T is tame of order 0.

Proposition 4.1. Suppose E is a locally convex space and T ⊂ L(E) is a finite set ofoperators. Then the following conditions are equivalent:4

(i) T is localizable;(ii) each operator T ∈ T is localizable;(iii) if ‖ · ‖ν : ν ∈ Λ is a directed system of prenorms defining the topology on E,

then for any T ∈ T and any ν ∈ Λ there are µ ∈ Λ and C > 0 such that‖Tnx‖ν ≤ Cn‖x‖µ for any x ∈ E and n ∈ Z+;

(iv) if ‖ · ‖ν : ν ∈ Λ is a directed system of prenorms defining the topology on E,then for any ν ∈ Λ there are µ ∈ Λ and C > 0 such that ‖Tnx‖ν ≤ Cn‖x‖µ forany T ∈ T, x ∈ E, and n ∈ Z+.

Proof. (i)=⇒(ii). This is obvious.(ii)=⇒(iii). Fix an arbitrary ν ∈ Λ. As T is localizable, it follows that there are T -

stable prenorms ‖ · ‖ on E, an element µ ∈ Λ, and a constant C1 ≥ 1 such that ‖ · ‖ν ≤‖ · ‖ ≤ C1‖ · ‖µ. Choose a C2 > 0 such that ‖Tx‖ ≤ C2‖x‖ for all x ∈ E. Then for anyx ∈ E and any n ∈ Z+ we have ‖Tnx‖ν ≤ ‖Tnx‖ ≤ Cn

2 ‖x‖ ≤ C1Cn2 ‖x‖µ. It remains to

set C = C1C2.(iii)=⇒(iv). This easily follows from the finiteness of T.(iv)=⇒(i). Let T = T1, . . . , Tk. For each finite set α = (α1, . . . , αs) of an arbitrary

length, where αi ∈ 1, . . . , k for each i, set |α| = s and Tα = Tα1 . . . Tαs. We also assume

that the identity operator corresponds to the set of zero length (s = 0) (cf. the notationof Example 3.2). Fix an arbitrary ν ∈ Λ and choose C > 0 and µ ∈ Λ satisfying (iii).Then ‖Tαx‖ν ≤ C |α|‖x‖µ for any set α of the above form and any x ∈ E. Therefore, forthe prenorm ‖ · ‖′ν on E defined by the formula

(4.2) ‖x‖′ν = supα

‖Tαx‖ν

C |α| ,

we have ‖ · ‖ν ≤ ‖ · ‖′ν ≤ ‖ · ‖µ. It now follows that the topology on E is defined by thesystem of prenorms ‖ · ‖′ν : ν ∈ Λ. On the other hand, it easily follows from (4.2) that‖Tix‖′ν ≤ C‖x‖′ν for any x ∈ E and any i = 1, . . . , k. Therefore, T is localizable.

Remark 4.1. Recall (see, for example, [72] or [57]) that a linear operator T in an LCS E iscalled regular if for some λ > 0 the set of operators (λT )n : n ∈ Z+ is equicontinuous.If the topology of E is defined by a system of prenorms ‖ · ‖ν : ν ∈ Λ, then T isregular if and only if there is a C > 0 such that for any ν ∈ Λ there is a µ ∈ Λsuch that ‖Tnx‖ν ≤ Cn‖x‖µ for any x ∈ E and any n ∈ Z+. Comparing this withcondition (iii) of the preceding proposition, we see that any regular operator is localizable.The converse is however not true; for example, it is not difficult to check that the operatorof multiplication by an independent variable in the space O(C) of entire functions islocalizable (because of the local m-convexity of O(C)) but not regular (since there is noλ > 0 such that the family of functions z → (λz)n : n ∈ Z+ is bounded in O(C)).

Suppose now that R is a ⊗-algebra, α : R → R is a (continuous) endomorphism,and δ : R → R is a (continuous) derivation. Assume in addition that α and δ arelocalizable. Our immediate goal is to show that on the space O(C, R) of R-valued entirefunctions there is a jointly continuous multiplication which coincides with multiplicationon R[z; α, δ] when it is restricted to polynomials. To this end, we first recall that O(C, R)is isomorphic (as an LCS and as a left R-⊗-module) to the projective tensor product

4In the case of one operator this proposition was independently proved by W. Zelazko [73].



R ⊗O(C) and that for any system of prenorms ‖ · ‖ν : ν ∈ Λ defining the topologyon R the topology on O(C, R) is defined by all prenorms of the form

(4.3) ‖f‖ν, ρ =∞∑

n=0

‖cn‖νρn (ν ∈ Λ, ρ > 0),

where f(z) =∑

n cnzn is the Taylor expansion of f ∈ O(C, R).

Lemma 4.2. Let R be a ⊗-algebra, α : R → R a localizable endomorphism, and δ : R → Ra localizable derivation. Then there is a unique continuous linear map

τ : O(C) ⊗R → R ⊗O(C),

such that

(4.4) τ (zn ⊗ r) =n∑

k=0

Sn,k(r) ⊗ zn−k

for all r ∈ R and n ∈ Z+.

Proof. The uniqueness is obvious. To prove the existence, we first define the map on theelements zn ⊗ r by (4.4) and extend it by linearity to polynomials. Since polynomialsform a dense subspace of O(C, R), it remains to check the continuity of the constructedmap.

By Proposition 4.1 (implication (ii)=⇒(i)), there is a system of α, δ-stable prenorms‖ ·‖ν : ν ∈ Λ defining the topology on R. Fix an arbitrary ν ∈ Λ and choose a constantC > 0 such that ‖α(r)‖ν ≤ C‖r‖ν and ‖δ(r)‖ν ≤ C‖r‖ν for any r ∈ R. Then for anyk, n ∈ Z+, k ≤ n, we have

(4.5) ‖Sn,k(r)‖ν ≤(

n

k

)Cn‖r‖ν .

For any ρ > 0, let ‖ · ‖ρ,ν be the prenorm on O(C) ⊗R corresponding to the prenorm‖·‖ν, ρ on R ⊗O(C) under the permutation isomorphism O(C) ⊗R ∼= R ⊗O(C). Then (4.4),(4.3), and (4.5) imply that

‖τ (zn ⊗ r)‖ν, ρ =∥∥∥∥ n∑

k=0

Sn,k(r) ⊗ zn−k

∥∥∥∥ν, ρ

=n∑

k=0

‖Sn,k(r)‖νρn−k

≤ Cn

(n∑

k=0

(n

k

)ρn−k

)‖r‖µ = Cn(ρ + 1)n‖r‖µ = ‖zn ⊗ r‖C(ρ+1),µ.

This, together with (4.3), imply that ‖τ (f)‖ν, ρ ≤ ‖f‖C(ρ+1),µ for any polynomial f =∑n zn⊗rn, and therefore τ is continuous on the subspace of polynomials, which is dense

in R ⊗O(C). Extending τ by continuity to O(C, R), we have the desired assertion.

Remark 4.2. If α = 1R, then the map τ from the preceding lemma can also be definedas follows. As δ is localizable, the formula

(4.6) f · r =∞∑

n=0

cnδn(r)(f ∈ O(C), f(z) =

∞∑n=0

cnzn, r ∈ R)

defines a structure of a left O(C)-⊗-module on R. Consider the maps

mR : O(C) ⊗R → R, f ⊗ r → f · r,∆: O(C) → O(C) ⊗O(C) ∼= O(C × C), (∆f)(z, w) = f(z + w).



It is not difficult to check that the composition

O(C) ⊗R∆⊗1R−−−−→ O(C) ⊗O(C) ⊗R

c23−−→ O(C) ⊗R ⊗O(C)mR⊗1O(C)−−−−−−−→ R ⊗O(C),

(4.7)

where c23 is the transposition of the 2-nd and 3-rd tensor factors, coincides with the map τfrom the previous lemma.

Remark 4.3. Suppose now that the endomorphism α is arbitrary and δ = 0. Replacingδ by α in (4.6), we have another structure of a left O(C)-⊗-module on R. Consider themap

∆′ : O(C) → O(C) ⊗O(C) ∼= O(C × C), (∆′f)(z, w) = f(zw).Now to get the map τ from the preceding lemma it suffices to replace ∆ by ∆′ in (4.7).

We now return to the general case and set A = R ⊗O(C). Define a multiplicationoperator mA : A ⊗A → A as the composition

A ⊗A = R ⊗O(C) ⊗R ⊗O(C)1R⊗τ⊗1O(C)−−−−−−−−→ R ⊗R ⊗O(C) ⊗O(C)

mR⊗mO(C)−−−−−−−→ R ⊗O(C) = A.(4.8)

Proposition 4.3. The operator mA : A ⊗A → A introduced above makes A a ⊗-algebra.Moreover the tautological embedding i : R[z; α, δ] → A is a homomorphism.

Proof. Combining (4.1), (4.4), and (4.8), we see that the diagram

(4.9)

R[z; α, δ]⊗R[z; α, δ] m

i⊗i

R[z; α, δ]

i

A ⊗A m

A

is commutative. This, together with the fact that the image of i is dense in A, impliesthat the multiplication on A is associative and that i is a homomorphism. Definition 4.3. The algebra A = R ⊗O(C) with the above multiplication will be de-noted O(C, R; α, δ) and called the analytic Ore extension of R. When α = 1R this algebrawill be denoted O(C, R; δ) and when δ = 0 it will be denoted O(C, R; α).

Notice that O(C, R; α, δ) contains R as a closed subalgebra and is therefore an R-⊗-algebra.

Remark 4.4. It follows from Remarks 4.2 and 4.3 that the algebras O(C, R; δ) andO(C, R; α) are particular cases of the analytic smash product of R and the ⊗-bialgebra Hacting on it (see [49]), the same way as R[t; δ] and R[t; α] are particular cases of the alge-braic smash product (see [65]). In the case of O(C, R; δ) (resp., R[t; δ]) for H one takesthe bialgebra (O(C), ∆) (resp., (C[t], ∆)), and in the case of O(C, R; α) (resp., R[t; α]) onetakes the bialgebra (O(C), ∆′) (resp., the bialgebra (C[t], ∆′) isomorphic to the semigroupbialgebra CZ+).

Remark 4.5. Notice that α and δ being localizable is not only a sufficient but alsoa necessary condition for the multiplication on R[z; α, δ] to extend to A = R ⊗O(C).Indeed, if m : A ⊗A → A is a continuous linear operator making diagram (4.9) commute,then the map

τ : O(C) ⊗R → R ⊗O(C), τ (a ⊗ r) = m(1 ⊗ a ⊗ r ⊗ 1)

must satisfy (4.4). Suppose ‖ ·‖ν : ν ∈ Λ is a system of prenorms defining the topologyon R. Fix an arbitrary ν ∈ Λ. It follows from the continuity of τ that there are µ ∈ Λ



and constants C, ρ > 0 such that ‖τ (u)‖ν, 1 ≤ C‖u‖ρ,µ for all u ∈ O(C) ⊗R. Thereforefor any r ∈ R and any n ∈ Z+ we have an estimate

‖δn(r)‖ν ≤n∑

k=0

‖Sn,k(r)‖ν =∥∥∥∥ n∑

k=0

Sn,k(r) ⊗ zn−k

∥∥∥∥ν, 1

= ‖τ (zn ⊗ r)‖ν, 1 ≤ C‖zn ⊗ r‖ρ,µ = Cρn‖r‖µ.

Taking account of Proposition 4.1 (implication (iii)=⇒(ii)) we see that δ is localizable.A similar argument shows that α is localizable.

Proposition 4.4. Suppose R is a ⊗-algebra, α : R → R is a localizable endomorphism,and δ : R → R is a localizable α-derivation. Suppose that B is an R-⊗-algebra which isalso Arens–Michael and that x ∈ B is such that x · r = α(r) ·x+ ηB(δ(r)) for any r ∈ R.Then there exists a unique continuous R-homomorphism f : O(C, R; α, δ) → B such thatf(z) = x.

Proof. For an arbitrary element a =∑

n rnzn ∈ O(C, R; α, δ) we set f(a) =∑

n rn · xn.Since B is an Arens–Michael algebra, it is easy to see that the latter series convergesabsolutely in B and that f : O(C, R; α, δ) → B is a continuous linear map. The factthat f is an R-homomorphism (and that f is a unique continuous R-homomorphismwith the required properties) easily follows from the universal property of the algebraR[z; α, δ] (see above) and the density of R[z; α, δ] in O(C, R; α, δ).

Next, we want to determine under what conditions the algebra O(C, R; δ) is Arens–Michael.

Definition 4.4. Suppose R is an Arens–Michael algebra. We shall say that a familyT ⊂ L(R) is m-localizable if the topology on R can be defined by a family of T-stablesubmultiplicative prenorms. We shall say that an operator T ∈ L(R) is m-localizable ifthe family T is m-localizable.

Example 4.1. It is easy to see that for any r in the Arens–Michael algebra R themultiplication operators Lr : s → rs and Rr : s → sr are m-localizable. Of course, thesame is true for their compositions and linear combinations and, in particular, for innerderivations and inner automorphisms of R.

Proposition 4.5. Let R be an Arens–Michael algebra, α : R → R an endomorphism,and δ : R → R an α-derivation. Suppose that the family α, δ is m-localizable. ThenO(C, R; α, δ) is an Arens–Michael algebra.

Proof. Suppose ‖ · ‖ν : ν ∈ Λ is a system of submultiplicative α, δ-stable prenormsdefining the topology on R. Fix an arbitrary ν ∈ Λ and choose a constant C > 0 suchthat ‖α(r)‖ν ≤ C‖r‖ν and ‖δ(r)‖ν ≤ C‖r‖ν for any r ∈ R. Then for any k, n ∈ Z+ andk ≤ n, inequality (4.5) holds.

We shall show that for any a, b ∈ A = O(C, R; α, δ) and any ρ > 0 we have

(4.10) ‖ab‖ν, ρ ≤ ‖a‖ν, C(ρ+1)‖b‖ν, ρ.

If a = s ⊗ zn and b = r ⊗ zm, then

‖s ⊗ zn · r ⊗ zm‖ν, ρ =∥∥∥∥ n∑

k=0

sSn,k(r) ⊗ zn+m−k

∥∥∥∥ν, ρ

=n∑

k=0

‖sSn,k(r)‖νρn+m−k

≤ ‖s‖νCn

(n∑

k=0

(n

k

)ρn−k

)‖r‖νρm = ‖s‖νCn(ρ + 1)n‖r‖νρm

= ‖s ⊗ zn‖ν, C(ρ+1)‖r ⊗ zm‖ν, ρ.



Thus, (4.10) holds for monomials. If a and b are polynomials, then they can be writtenas finite sums a =

∑i ai and b =

∑j bj , where ai and bj are monomials; moreover,

‖a‖ν, ρ =∑

i ‖ai‖ν, ρ and ‖b‖ν, ρ =∑

j ‖bj‖ν, ρ for any ν ∈ Λ and any ρ > 0. This impliesthat

‖ab‖ν, ρ =∥∥∥∑

i,j

aibj

∥∥∥ν, ρ

≤∑i,j

‖ai‖ν, C(ρ+1)‖bj‖ν, ρ

=(∑

i

‖ai‖ν, C(ρ+1)

)(∑j

‖bj‖ν, ρ

)= ‖a‖ν, C(ρ+1)‖b‖ν, ρ.

Therefore, (4.10) holds for all polynomials and therefore for arbitrary a, b ∈ A. UsingCorollary 2.2, we conclude that A is Arens–Michael.

Remark 4.6. Suppose R, α, δ are the same as in the preceding proposition. Then theproperty of A = O(C, R; α, δ) established in Proposition 4.4 is obviously universal andcompletely characterizes the pair (A, z ∈ A). In other words, if R-AM denotes thecategory of R-⊗-algebras which are Arens–Michael, then we have a bijection

Hom R-AM(O(C, R; α, δ), B) ∼=x ∈ B : x · r = α(r) · x + ηB(δ(r)) ∀r ∈ R

,

giving rise to an isomorphism of functors of B.

Remark 4.7. The author does not know if an analog of Proposition 4.1 holds for m-localizable families and if there are localizable families of the form α, δ which are notm-localizable.

We remark that the prenorms (4.3) defining the topology on O(C, R; α, δ) are not nec-essarily submultiplicative, even in the case when O(C, R; α, δ) is Arens–Michael (cf. (4.10)).Thus it is convenient to single out a special case when the norms are submultiplicative.

Definition 4.5. Suppose R is an Arens–Michael algebra. An operator T ∈ L(R) issaid to be m-contracting (resp., m-isometric) if the topology on R can be defined by afamily ‖ · ‖ν : ν ∈ Λ of submultiplicative prenorms such that ‖Tr‖ν ≤ ‖r‖ν (resp.,‖Tr‖ν = ‖r‖ν) for all r ∈ R.

Proposition 4.6. Suppose R is an Arens–Michael algebra and α : R → R is an m-contracting endomorphism. Fix a system of submultiplicative prenorms ‖ · ‖ν : ν ∈ Λdefining the topology on R and such that ‖α(r)‖ν ≤ ‖r‖ν for all r ∈ R and ν ∈ Λ. Theneach of the prenorms (4.3) on O(C, R; α) is submultiplicative.

Proof. For any s, r ∈ R we have

‖s ⊗ zn · r ⊗ zm‖ν, ρ = ‖sαn(r) ⊗ zn+m‖ν, ρ

≤ ‖s‖ν‖r‖νρn+m = ‖s ⊗ zn‖ν, ρ‖r ⊗ zm‖ν, ρ.

This implies, as in the proof of Proposition 4.5, that ‖ab‖ν, ρ ≤ ‖a‖ν, ρ‖b‖ν, ρ for anya, b ∈ O(C, R; α).

Remark 4.8. If α = 1R, then it is not difficult to show that the conditions of Propo-sition 4.5 are not only sufficient but also necessary for O(C, R; δ) to be Arens–Michael.Indeed, if O(C, R; δ) is Arens–Michael, then so is its closed subalgebra R. Furthermore,the derivation δ coincides with the restriction to R of the inner derivation adz and there-fore is m-localizable (see Example 4.1).

Similarly, if δ = 0, α is an automorphism, and O(C, R; α) is Arens–Michael, then α,being the restriction to R of the inner automorphism Adz, is m-localizable.



Finally, if δ = 0, ‖1‖ν = 1 for all ν ∈ Λ, and each of the prenorms (4.3) on O(C, R; α)is submultiplicative, then α is necessarily m-contracting. This follows at once from thechain of inequalities

‖α(r)‖ν = ‖α(r)z‖ν, 1 = ‖zr‖ν, 1 ≤ ‖z‖ν, 1‖r‖ν, 1 = ‖r‖ν .

4.2. Analytic tensor algebras. Suppose R is an arbitrary associative algebra (withouttopology) and M is an R-bimodule. For any n ∈ N set

(4.11) M⊗n = M ⊗R · · · ⊗R M︸︷︷︸n

.

We also set M⊗0 = R. Recall that the tensor algebra of M is defined as the R-bimodule

TR(M) =∞⊕

n=0

M⊗n

together with the multiplication uniquely determined by the rule

ab = a ⊗ b for a ∈ M⊗m, b ∈ M⊗n

(here we used the canonical isomorphisms M⊗m ⊗R M⊗n ∼= M⊗(m+n) that hold for anym, n ∈ Z+).

For any n ∈ Z+ let in : M⊗n → TR(M) denote the canonical embedding. Clearly, inis a morphism of R-bimodules and also i0 is an algebra homomorphism making TR(M)an R-algebra.

The tensor algebra can also be defined as a pair (TR(M), i1) consisting of an R-algebraTR(M) and an R-bimodule morphism i1 : M → TR(M) such that for any R-algebra A andany R-bimodule morphism α : M → A there exists a unique R-algebra homomorphismψ : TR(M) → A making the following diagram commute:

TR(M)ψ A

M

i1

α

Example 4.2. Suppose R is an algebra and α : R → R is an endomorphism. Thenthe Ore extension R[t; α] is canonically isomorphic to the tensor algebra TR(M), whereM = Rα (see Section 2); under this isomorphism 1 ∈ M corresponds to t ∈ R[t; α].

Later we shall describe an “analytic analog” of the tensor algebra, the analytic tensoralgebra TR(M), which is a generalization of Taylor’s algebra of power series in severalfree variables (see Example 3.2) and also of Cuntz’s “smooth tensor algebra” [8, 9]. Weshall then show that the Arens–Michael envelope of the tensor algebra is an analytictensor algebra.

Suppose R is a ⊗-algebra and M is an R-⊗-bimodule. Replacing the algebraic tensorproduct ⊗R in (4.11) by the projective tensor product ⊗R (see Section 1) we have afamily of R-⊗-bimodules M ⊗n (n ∈ Z+). Suppose ‖ · ‖ν : ν ∈ Λ is a directed systemof prenorms defining the topology on M. For any ν ∈ Λ and any n ∈ N , let ‖ · ‖⊗n

ν

denote the prenorm on M ⊗n which is the projective tensor product of n copies of ‖ · ‖ν .Consider the space TR(M)+ consisting of all x = (xn) ∈

∏∞n=1 M ⊗n such that

‖x‖ν, ρ =∞∑

n=1

‖xn‖⊗nν ρn < ∞ ∀ρ > 0.



The system of prenorms ‖ · ‖ν, ρ : ν ∈ Λ, ρ > 0 makes TR(M)+ an LCS which is, asis easily seen, complete. It is also easy to show that the topology on TR(M)+ does notdepend on the defining system ‖ · ‖ν : ν ∈ Λ of prenorms on M.

We now setTR(M) = R ⊕ TR(M)+.

The elements of TR(M) which belong to M ⊗n for some n ∈ Z+ will be called homoge-neous. We shall use a similar convention for TR(M).

Consider the canonical map i : TR(M) → TR(M) sending each elementary tensorx1 ⊗ · · · ⊗ xn ∈ M⊗n to x1 ⊗ · · · ⊗ xn ∈ M ⊗n. It is easy to see that its image is densein TR(M).

Proposition 4.7. There is a unique multiplication on TR(M) making it a ⊗-algebraand such that i is a homomorphism. On the homogeneous elements this multiplicationcoincides with the canonical map

(4.12) M ⊗m × M ⊗n → M ⊗(m+n), (a, b) → a ⊗ b.

Moreover TR(M)+ is an ideal of TR(M) and a (nonunital) Arens–Michael algebra.If in addition R is an Arens–Michael algebra, then so is TR(M).

Proof. Let TR(M)fin denote the subspace of TR(M) consisting of compactly supportedsequences. Clearly, there exists a unique bilinear operator

(4.13) TR(M)fin × TR(M)fin → TR(M)fin, (x, y) → xy ,

coinciding on homogeneous elements with canonical map (4.12). Let x = (xn)n≥1 andy = (yn)n≥1 be elements of TR(M)fin ∩ TR(M)+. Then for any ν ∈ Λ and any ρ > 0 wehave:

‖xy‖ν, ρ =∑

n

∥∥ ∑i+j=n

xiyj

∥∥⊗n

νρn

≤∑

n

∑i+j=n

‖xiyj‖⊗nν ρn ≤

∑n

∑i+j=n

‖xi‖⊗iν ‖yj‖⊗j

ν ρn

=(∑

i

‖xi‖⊗iν ρi)(∑

j

‖yj‖⊗jν ρj)

= ‖x‖ν, ρ‖y‖ν, ρ.

(4.14)

Furthermore, as the action of R on M is jointly continuous, we have that for any ν ∈ Λthere are µ ∈ Λ and a continuous prenorm ‖ · ‖ on R such that

‖a · x‖ν ≤ ‖a‖‖x‖µ and ‖x · a‖ν ≤ ‖x‖µ‖a‖ (a ∈ R, x ∈ M).

We may assume that ‖ · ‖µ majorizes ‖ · ‖ν . This, together with the properties of theprojective tensor product, imply that

‖a · x‖⊗nν ≤ ‖a‖‖x‖⊗n

µ and ‖x · a‖⊗nν ≤ ‖x‖⊗n

µ ‖a‖

(a ∈ R, x ∈ M ⊗n, n ∈ N ),

whence

(4.15) ‖a · x‖ν, ρ ≤ ‖a‖‖x‖µ,ρ and ‖x · a‖ν, ρ ≤ ‖x‖µ,ρ‖a‖ (a ∈ R, x ∈ TR(M)+).

As the multiplication on R is jointly continuous, (4.14) and (4.15) show that the bilinearoperator (4.13) is also jointly continuous. Therefore, it extends to a jointly continuous



bilinear operator mT : TR(M)× TR(M) → TR(M). Let mT denote the multiplication onTR(M); then it follows easily from the construction that the diagram

TR(M) × TR(M)mT

i×i

TR(M)

i

TR(M) × TR(M)

mT TR(M)

is commutative. Since the image of i is dense in TR(M), we see that mT is associative(hence TR(M) is a ⊗-algebra), and that i is a homomorphism. Thus we have a multi-plication on TR(M) with the required properties. Its uniqueness is obvious because theimage of i is dense. That TR(M)+ is a closed ideal of TR(M) is clear from the construc-tion. Finally, it follows from (4.14) that TR(M)+ is Arens–Michael. This proves the firstpart of the proposition.

Assume now that R is an Arens–Michael algebra, and choose a directed system ofsubmultiplicative prenorms ‖ · ‖λ : λ ∈ Λ1 defining the topology on R. By Proposi-tion 3.4 (see also Remark 3.4), without loss of generality we may assume that for anyν ∈ Λ the prenorm ‖ · ‖ν on M is m-compatible. As in the proof of Proposition 3.6, witheach ν ∈ Λ we associate ϕ(ν) ∈ Λ1 such that

‖a · x‖ν ≤ ‖a‖ϕ(ν)‖x‖ν and ‖x · a‖ν ≤ ‖x‖ν‖a‖ϕ(ν) (a ∈ R, x ∈ M),

and set Λ′ = (λ, ν) ∈ Λ1 × Λ : ϕ(ν) ≺ λ. As in (4.15), it follows from the last formulathat

(4.16) ‖a · x‖ν, ρ ≤ ‖a‖λ‖x‖ν, ρ and ‖x · a‖ν, ρ ≤ ‖x‖ν, ρ‖a‖λ (a ∈ R, x ∈ TR(M)+)

for any (λ, ν) ∈ Λ′ and any ρ > 0. Now we write an arbitrary element x = (xn) ∈ TR(M)in the form x = (x0, x), where x = (xn)n≥1 ∈ TR(M)+, and for any (λ, ν) ∈ Λ′ and anyρ > 0 define a prenorm ‖ · ‖λ, ν, ρ on TR(M) by the formula

(4.17) ‖x‖λ, ν, ρ = ‖x0‖λ + ‖x‖ν, ρ.

As in the proof of Proposition 3.6, we have that the topology on TR(M) is defined bythe system of prenorms of the form (4.17). On the other hand, it follows from (4.14) and(4.16) that

‖xy‖λ, ν, ρ = ‖x0y0‖λ + ‖x0y + xy0 + xy‖ν, ρ

≤ ‖x0‖λ‖y0‖λ + ‖x0‖λ‖y‖ν, ρ + ‖x‖ν, ρ‖y0‖λ + ‖x‖ν, ρ‖y‖ν, ρ

= (‖x0‖λ + ‖x‖ν, ρ)(‖y0‖λ + ‖y‖ν, ρ) = ‖x‖λ, ν, ρ‖y‖λ, ν, ρ

for any x, y ∈ TR(M), any (λ, ν) ∈ Λ′, and any ρ > 0. Therefore, all prenorms ‖·‖λ, ν, ρ :(λ, ν) ∈ Λ′, ρ > 0 are submultiplicative and TR(M) is Arens–Michael.

Definition 4.6. The ⊗-algebra TR(M) thus constructed will be called the analytic tensoralgebra of M.

Remark 4.9. When A = C the algebra TC(M) coincides with Cuntz’s “smooth tensoralgebra” [8]. If in addition n = dim M < ∞, we recover Taylor’s algebra Fn of powerseries in n free variables (see Example 3.2).

Now suppose, as before, that R is a ⊗-algebra and M is an R-⊗-bimodule. As in thecase of the usual tensor algebra TR(M), we have the canonical morphisms in : M ⊗n →TR(M) (n ∈ Z+) of R-⊗-bimodules; moreover i0 : R → TR(M) is an algebra homomor-phism making TR(M) an R-⊗-algebra.



Proposition 4.8. Suppose R is a ⊗-algebra and M is an R-⊗-bimodule. Then for anyR-⊗-algebra A which is Arens–Michael, and any morphism of R-⊗-bimodules α : M → A

there exists a unique continuous R-homomorphism ψ : TR(M) → A making the followingdiagram commute:

(4.18)

TR(M)ψ A

M

i1

α

Proof. We fix a directed system of prenorms ‖ · ‖ν : ν ∈ Λ defining the topology on Mand also choose a system of submultiplicative prenorms ‖ · ‖µ : µ ∈ Λ2 defining thetopology on A. For any n ∈ N consider the morphism

mnA : A⊗n → A, a1 ⊗ · · · ⊗ an → a1 · · · an.

As the prenorms ‖ · ‖µ are submultiplicative, we have

(4.19) ‖mnA(u)‖µ ≤ ‖u‖⊗n

µ for all u ∈ A⊗n, n ∈ N , µ ∈ Λ2.

Define a map ψ : TR(M) → A by the formula

ψ(x) = ηA(x0) +∞∑

n=1

(mnA α⊗n)(xn).

To show that ψ is well-defined and continuous, we fix an arbitrary µ ∈ Λ2 and, usingthe continuity of α, choose ν ∈ Λ and C > 0 such that ‖α(x)‖µ ≤ C‖x‖ν for all x ∈ M.Then, obviously,

(4.20) ‖α⊗n(xn)‖⊗nµ ≤ Cn‖xn‖⊗n

ν for all xn ∈ M ⊗n, n ∈ N .

Using (4.19) and (4.20), for each x ∈ TR(M) we have

‖ψ(x)‖µ ≤ ‖ηA(x0)‖µ +∞∑

n=1

‖(mnA α⊗n)(xn)‖µ

≤ ‖ηA(x0)‖µ +∞∑

n=1

Cn‖xn‖⊗nν = ‖ηA(x0)‖µ + ‖x‖ν, C .

Since ηA is continuous, the latter estimate implies that ψ is well-defined and continuous.The commutativity of diagram (4.18) follows directly from the construction. A simplecalculation on elementary tensors (which we omit) shows that ψ is a homomorphism ofR-algebras. Finally, the uniqueness of ψ with the required properties follows easily fromthe fact that the images of i0 and i1 generate a dense subalgebra of TR(M).

Suppose R is a ⊗-algebra and α is a (continuous) endomorphism of it. We shall showthat the analytic tensor algebra TR(Rα) can be realized as some algebra of “skew powerseries”. For this we fix a directed system of prenorms ‖ · ‖ν : ν ∈ Λ defining thetopology on R, and for each ν ∈ Λ and each n ∈ N define a prenorm ‖ · ‖(n)

ν on R by theformula

(4.21) ‖r‖(n)ν = inf

k∑i=1

‖r0,i‖ν · · · ‖rn−1,i‖ν ,



where inf is taken over all representations of r in the form

(4.22) r =k∑

i=1

r0,i α(r1,i)α2(r2,i) · · ·αn−1(rn−1,i) (rj,i ∈ R, k ∈ N ).

We also set ‖ · ‖(0)ν = ‖ · ‖ν . It is not difficult to see that for a fixed n the system of

prenorms ‖ · ‖(n)ν : ν ∈ Λ on R is equivalent to the original one. Consider the space

Rt; α consisting of all formal expressions a =∑∞

n=0 rntn such that

‖a‖ν, ρ =∞∑

n=0

‖rn‖(n)ν ρn < ∞ ∀ρ > 0.

The system of prenorms ‖ · ‖ν, ρ : ν ∈ Λ, ρ > 0 makes Rt; α an LCS, which, as iseasy to see, is complete.

Proposition 4.9. There is a unique structure of a ⊗-algebra on Rt; α such that theembedding R[t; α] → Rt; α is a homomorphism. Moreover, there exists a unique R-⊗-algebra isomorphism Rt; α → TR(Rα) sending t to 1 ∈ Rα. Consequently, if R is anArens–Michael algebra, then so is Rt; α.

Proof. It is easy to see that for any n ∈ N we have an isomorphism of R-⊗-bimodules

ϕn : (Rα)⊗n → Rαn , r0 ⊗ · · · ⊗ rn−1 → r0 α(r1) · · ·αn−1(rn−1).

Moreover,‖ϕn(u)‖(n)

ν = ‖u‖⊗nν

for any ν ∈ Λ and any u ∈ (Rα)⊗n. Therefore, there is an isomorphism of LCS,

Rt; α → TR(Rα),∞∑

n=0

rntn → x = (xn)∞n=0,

where xn = rn ⊗ 1 ⊗ · · · ⊗ 1︸︷︷︸n−1

∈ (Rα)⊗n.

It remains to carry the structure of a ⊗-algebra from TR(Rα) to Rt; α. Under this con-struction, if R[t; α] is identified with TR(Rα) (see Example 4.2), then the homomorphismi : TR(Rα) → TR(Rα) corresponds to R[t; α] → Rt; α (see Proposition 4.7). Therefore,the above embedding is an algebra homomorphism. The assertion about uniqueness ob-viously follows from the density of R[t; α] in Rt; α. Finally, if R is an Arens–Michaelalgebra, then so is TR(Rα) (see Proposition 4.7) and therefore Rt; α.

Combining the preceding proposition with Proposition 4.8, we have the followingproperty of Rt; α.

Corollary 4.10. Suppose R is a ⊗-algebra and α : R → R is an endomorphism. Thenfor any R-⊗-algebra A which is Arens–Michael and any x ∈ A such that x · r = α(r) · xfor all r ∈ R there is a unique continuous R-homomorphism ψ : Rt; α → A such thatψ(t) = x.

Remark 4.10. If R is an Arens–Michael algebra and M is an R-⊗-bimodule, then theproperty of TR(M) established in Proposition 4.8 is obviously universal and completelycharacterizes the pair (TR(M), i1). In other words, there is a bijection

Hom R-AM(TR(M), A) ∼= RhR(M, A),



giving rise to an isomorphism of functors of A. In particular, if α is an endomorphismof R, then

Hom R-AM(Rt; α, A) ∼=x ∈ A : x · r = α(r) · x ∀r ∈ R

(cf. Remark 4.6).

Proposition 4.11. Assume, in addition to the conditions of Corollary 4.10 that R is anArens–Michael algebra and α is m-localizable. Then there exists a unique R-isomorphism

Rt; α → O(C, R; α), t → (z → z · 1).

Proof. This is a consequence of the universal properties of the above algebras (see Re-marks 4.6 and 4.10).

Example 4.3. We shall now give an example of an endomorphism α of an Arens–Michael algebra R such that the underlying LCS of Rt; α coincides with the spaceof formal power series R[[t]]. Of course, in view of the preceding proposition, such anendomorphism cannot be m-localizable.

Set R = O(C) and define α : R → R by setting α(f)(z) = f(z − 1) (z ∈ C). For thedefining system of prenorms on R take ‖ · ‖ρ : ρ > 0, where ‖f‖ρ = max|z|≤ρ |f(z)|.We shall show that for any ρ > 0 there is an N ∈ N such that ‖ · ‖(n)

ρ ≡ 0 for all n > N .Henceforth, for z ∈ C and r > 0, the open disc of radius r and center z will be denotedUr(z).

Fix an arbitrary ρ > 0 and choose an r > ρ such that 2r < [2ρ]+1. Then for any integern ≥ [2ρ] + 1 we have Ur(−n) ∩ Ur(0) = ∅. Set U = Ur(−n) ∪ Ur(0), take an arbitraryf ∈ O(C) and ε > 0, and define a function h ∈ O(U) by setting h(z) = 0 for z ∈ Ur(0)and h(z) = f(z + n) for z ∈ Ur(−n). By Runge’s theorem (see, for example, [42, Ch. 4,Section 2]), the restriction homomorphism O(C) → O(U) has a dense image. Hencethere is a g ∈ O(C) such that |g(z) − h(z)| < ε/2 for any z ∈ Uρ(−n) ∪ Uρ(0). It followsfrom the equality f = αn(g) + (f − αn(g)) and the definition of the prenorms ‖ · ‖(n)

ρ

(see (4.21) and (4.22)) that

(4.23) ‖f‖(n+1)ρ ≤ ‖g‖ρ + ‖f − αn(g)‖ρ.

By the choice of h and g, we have ‖g‖ρ < ε/2. Furthermore,

‖f − αn(g)‖ρ = max|f(z) − g(z − n)| : z ∈ Uρ(0)

= max

|f(z + n) − g(z)| : z ∈ Uρ(−n)

= max

|h(z) − g(z)| : z ∈ Uρ(−n)

<

ε

2.

Comparing this with (4.23), we have an estimate ‖f‖(n+1)ρ < ε. As ε > 0 and f ∈ O(C)

were arbitrary, this means that ‖ · ‖(n+1)ρ ≡ 0 for all n ≥ [2ρ] + 1. Therefore it follows

from the definition of Rt; α that, as a locally convex space, it coincides with the spaceof formal power series R[[t]].

4.3. Analytic skew Laurent extensions. Suppose R is an arbitrary associative alge-bra (without topology) and α : R → R is an automorphism. Recall that the skew Laurentextension R[t, t−1; α], as a left R-module, consists of all Laurent polynomials

∑ni=−m rit

i

in t with coefficients in R; the multiplication on R[t, t−1; α] is uniquely determined bythe relation tr = α(r)t for r ∈ R and the requirement that the natural embeddingsR → R[t, t−1; α] and C[t, t−1] → R[t, t−1; α] be algebra homomorphisms. Notice thatthe former embedding makes R[t, t−1; α] an R-algebra. The algebra R[t, t−1; α] and its



element t have the following universal property, which completely characterizes them:For any R-algebra A and any invertible element x ∈ A such that x · r = α(r) · x (r ∈ R),there exists a unique R-homomorphism f : R[t, t−1; α] → A such that f(t) = x. Clearly,R[t, t−1; α] contains R[t; α] as a subalgebra.

Now we shall construct a locally convex analog of the algebra R[t, t−1; α]. Suppose R isa ⊗-algebra. Set C× = C\0 and recall that for any system of prenorms ‖ ·‖ν : ν ∈ Λdefining the topology on R, the topology on O(C×, R) is defined by all prenorms of theform

(4.24) ‖f‖ν, ρ =∞∑

n=−∞‖cn‖νρn (ν ∈ Λ, ρ > 0),

where f(z) =∑

n cnzn is the Laurent expansion of f ∈ O(C×, R).

Lemma 4.12. Suppose R is a ⊗-algebra and α : R → R is an automorphism. If α andα−1 are localizable, then there is a unique continuous linear map

τ : O(C×) ⊗R → R ⊗O(C×),

such that

(4.25) τ (zn ⊗ r) = αn(r) ⊗ zn

for all r ∈ R and n ∈ Z.

The proof of this lemma is similar and in fact simpler than that of Lemma 4.2 and istherefore omitted. Notice that the description of τ given in Remark 4.3 carries over inan obvious way to the case in question.

Now let A = R ⊗O(C×) and define a multiplication mA : A ⊗A → A by the formulaobtained from (4.8) by replacing C with C×. It is easy to see that we have the followinganalog of Proposition 4.3.

Proposition 4.13. The operator mA : A ⊗A → A thus defined makes A a ⊗-algebra.Moreover the tautological embedding i : R[z, z−1; α] → A is a homomorphism.

Definition 4.7. The algebra A = R ⊗O(C×) with the above multiplication will bedenoted O(C×, R; α) and called the analytic Laurent extension of R.

Notice that the algebra O(C×, R; α) contains R as a closed subalgebra and is thereforean R-⊗-algebra. As in the case of analytic Ore extensions (see Remark 4.4), this algebracan be described as the analytic smash product of R and the ⊗-bialgebra (O(C×), ∆′)acting on it (the latter, as is not difficult to see, is the Arens–Michael envelope of thegroup bialgebra CZ). Finally, similarly to the argument in Remark 4.5, one shows thatα and α−1 being localizable is not only a sufficient but also a necessary condition for themultiplication on R[z, z−1; α] to extend to A = R ⊗O(C).

The next three propositions are proved similarly to Propositions 4.4, 4.5, and 4.6. Weomit their proofs.

Proposition 4.14. Let R be a ⊗-algebra and α : R → R an automorphism such that αand α−1 are localizable. Suppose that B is an R-⊗-algebra which is Arens–Michael andx ∈ B is an invertible element such that x · r = α(r) · x for any r ∈ R. Then there existsa unique continuous R-homomorphism

f : O(C×, R; α) → B,

such that f(z) = x.



Proposition 4.15. Let R be an Arens–Michael algebra and α : R → R an automorphism.Suppose that the family α, α−1 is m-localizable. Then O(C×, R; α) is an Arens–Michaelalgebra.

Proposition 4.16. Let R be an Arens–Michael algebra and α : R → R an m-isometricautomorphism. Fix a system of submultiplicative prenorms ‖ · ‖ν : ν ∈ Λ defining thetopology on R and such that ‖α(r)‖ν = ‖r‖ν for all r ∈ R and ν ∈ Λ. Then each of theprenorms (4.24) on O(C×, R; α) is submultiplicative.

Notice that the equality α±1 = Adz±1 immediately implies that α, α−1 being m-localizable is not only sufficient but also necessary for O(C×, R; α) to be Arens–Michael.If ‖1‖ν = 1 for all ν and each of the prenorms (4.24) on O(C×, R; α) is submultiplicative,then α must be m-isometric (cf. Remark 4.8).

In the case of the analytic Laurent extension, the universal property of the analyticOre extension mentioned in Remark 4.6 becomes

(4.26) Hom R-AM(O(C×, R; α), B) ∼=x ∈ B : x is invertible, x · r = α(r) · x ∀r ∈ R

.

5. Examples of Arens–Michael envelopes

5.1. Ore extensions: the case α = 1R. Universal enveloping algebra. Suppose Ris an algebra (without topology) and δ : R → R is a derivation. Let Rδ be the Arens–Michael algebra obtained by completing R with respect to the system of all δ-stablesubmultiplicative prenorms. The canonical homomorphism R → Rδ will be denoted j.Clearly, δ uniquely defines an m-localizable derivation δ of the algebra Rδ such thatδ j = j δ. Moreover we have homomorphisms

R[z; δ] → Rδ[z; δ] → O(C, Rδ; δ) ,

where the former coincides with j on R and sends z to z, and the latter is the canonicalembedding (see above). The composition of these homomorphisms will be denoted ιR[z;δ].

Theorem 5.1. The pair (O(C, Rδ; δ), ι) is the Arens–Michael envelope of R[z; δ].

Proof. Let ϕ : R[z; δ] → B be a homomorphism from R[z; δ] to an Arens–Michael alge-bra B. Set f = ϕ|R and x = ϕ(z). Fix a submultiplicative prenorm ‖ · ‖ on B anddefine a prenorm ‖ · ‖f on R by the formula ‖r‖f = ‖f(r)‖. Clearly, this prenorm issubmultiplicative and it is not difficult to check that it is δ-stable. Indeed,

‖δ(r)‖f =∥∥f(δ(r))

∥∥ =∥∥ϕ([z, r])

∥∥ =∥∥[x, f(r)]

∥∥ ≤ 2‖x‖ · ‖f(r)‖ = 2‖x‖ · ‖r‖f

for any r ∈ R. It now follows that f is continuous in the topology on R defined bythe system of all δ-stable submultiplicative prenorms. Therefore, there exists a uniquecontinuous homomorphism ηB : Rδ → B such that ηBj = f . Thus B becomes an Rδ-⊗-algebra and ϕ an R-Rδ-homomorphism. This implies that the relation x · r = r · x +ηB(δ(r)) holds for any r ∈ Im j and hence for any r ∈ Rδ. Therefore, by Proposition 4.4,there exists a unique continuous Rδ-homomorphism ϕ : O(C, Rδ; δ) → B such that ϕ(z) =x. Therefore, the composition ϕ ιR[z;δ] : R[z; δ] → B is an R-homomorphism andsends z to x. The uniqueness of an R-homomorphism with this property implies thatϕ ιR[z;δ] = ϕ, i.e., ϕ extends ϕ in the sense of Definition 3.1. The uniqueness of suchan extension is clear because the image of ιR[z;δ] is dense in O(C, Rδ; δ). The rest isclear.

Remark 5.1. A similar result can be proved for more general smash products (see [49],Theorem 2.2).



The above construction of the Arens–Michael envelope of R[t; δ] can be used, in partic-ular, for an inductive description of the Arens–Michael envelopes of universal envelopingalgebras of solvable Lie algebras. If g is a solvable finite-dimensional Lie algebra, then ithas an ideal h of codimension 1. Choose an arbitrary x ∈ g \ h and set R = U(h). It iseasy to see that U(g) ∼= R[x; δ], where δ = adx. Therefore to describe the Arens–Michaelenvelope of U(g) it suffices to know the system of all submultiplicative δ-stable prenormson U(h). We shall illustrate this approach by two very simple examples.

Example 5.1. Suppose g is a two-dimensional Lie algebra with basis x, y and commu-tator [x, y] = y. Set h = Cy and R = U(h) = C[y]. Consider the derivation δ = adx = y d

dy

of R; by the above, U(g) = R[x; δ] and our goal is to describe all submultiplicative δ-stable prenorms on R.

For any n ∈ Z+ and any r =∑

i ciyi ∈ R we set ‖r‖n =

∑ni=0 |ci|. A standard

calculation (which we omit) shows that ‖ · ‖n is a submultiplicative prenorm on R. Sinceδ(r) =

∑i iciy

i, we have ‖δ(r)‖n ≤ n‖r‖n for any r ∈ R, so that ‖ · ‖n is δ-stable.Moreover, any δ-stable prenorm on R is majorized by some ‖ ·‖n. Indeed, suppose ‖ ·‖ isa δ-stable prenorm and let a constant C > 0 be such that ‖δ(r)‖ ≤ C‖r‖ for any r ∈ R.Then for r = yn we have δ(yn) = nyn, whence n‖yn‖ ≤ C‖yn‖. Therefore, ‖yn‖ = 0 forsufficiently large n, which easily implies that ‖ · ‖ ≤ M‖ · ‖n for some n ∈ Z+ and someconstant M > 0. Using the notation introduced above, we have that Rδ coincides withthe completion of R with respect to the system of prenorms ‖ · ‖n : n ∈ Z+, i.e., withthe algebra of formal power series C[[y]]. Applying Theorem 5.1, we have the following

Proposition 5.2. Suppose g is a Lie algebra with basis x, y and commutator [x, y] = y.The Arens–Michael envelope U(g) of U(g) consists of all power series a =

∑i,j cijy

ixj

such that∑

j |cij |ρj < ∞ for any i ∈ Z+ and any ρ > 0. The topology on U(g) is definedby the system of prenorms

‖a‖n,ρ =∞∑

j=0

n∑i=0

|cij |ρj (n ∈ Z+, ρ > 0).

In other words, the underlying LCS of U(g) coincides with O(C)[[y]] = C[[y]] ⊗O(C),and the multiplication is uniquely determined by the relation [x, y] = y (where x ∈ O(C)denotes the identity map on C) and by the condition that the embeddings C[[y]] → U(g)and O(C) → U(g) be homomorphisms.

Example 5.2. Suppose g is a three-dimensional Lie algebra with basis x, y, z and com-mutator [x, y] = z, [x, z] = [y, z] = 0 (i.e., the Heisenberg algebra). Set h = spany, zand R = U(h) = C[y, z]. Consider the derivation δ = adx = z ∂

∂y of R. As in thepreceding example, we need to describe all submultiplicative δ-stable prenorms on R.

For any ρ > 0 and any a =∑

i,j aijyizj ∈ R we set

(5.1) ‖a‖ρ =∑i,j

|aij |i!

(i + j)!ρi+j .

Clearly, ‖ · ‖ρ is a norm on R.

Lemma 5.3. The norms (5.1) are submultiplicative and δ-stable. Moreover, any sub-multiplicative δ-stable prenorm on R is majorized by some norm (5.1).

Proof. Fix an arbitrary ρ > 0 and for each pair of indices (i, j) set pij = i!(i+j)!ρ

i+j .Obviously, to prove that ‖ · ‖ρ is submultiplicative it suffices to show that

pi+k,j+l ≤ pijpkl for any i, j, k, l ∈ Z+.



After canceling ρi+j+k+l, the last inequality becomes

(i + k)!(i + k + j + l)!

≤ i!k!(i + j)!(k + l)!

or, equivalently, (i + k

i

)≤(

i + k + j + l

i + j

).

This formula can easily be deduced by induction on j and l from the obvious inequalities(n

m

)≤(

n + 1m + 1

)and

(n

m

)≤(

n + 1m

).

Thus the norms (5.1) are submultiplicative.Furthermore, for any a =

∑i,j aijy

izj ∈ R we have

‖δ(a)‖ρ =∥∥∥∥∑

i,ji≥1

aijiyi−1zj+1

∥∥∥∥ρ

=∑i,ji≥1

|aij |i(i − 1)!(i + j)!

ρi+j ≤∑i,j

|aij |i!

(i + j)!ρi+j = ‖a‖ρ.

Therefore, the norms (5.1) are δ-stable.Now suppose that ‖ · ‖ is a submultiplicative δ-stable prenorm on R. Fix a constant

C ≥ 1 such that ‖δ(a)‖ ≤ C‖a‖ for any a ∈ R and notice that for any i, j ∈ Z+,

δj(yi+j) =(i + j)!

i!yizj .

Therefore, substituting a = yi+j in ‖δj(a)‖ ≤ Cj‖a‖, we have

(i + j)!i!

‖yizj‖ ≤ Cj‖yi+j‖ ≤ Cjri+j ,

where r = ‖y‖. Hence

‖yizj‖ ≤ Cj i!(i + j)!

ri+j ≤ i!(i + j)!

(Cr)i+j = ‖yizj‖Cr.

Therefore, setting ρ = Cr, for any a =∑

i,j aijyizj we have an estimate

‖a‖ =∥∥∥∑

i,j

aijyizj∥∥∥ ≤∑

i,j

|aij | ‖yizj‖ ≤∑i,j

|aij | ‖yizj‖ρ = ‖a‖ρ.

Thus ‖a‖ ≤ ‖a‖ρ for any a ∈ R.

Combining the last lemma with Theorem 5.1, we have

Proposition 5.4. Suppose g is a Lie algebra with basis x, y, z and commutator [x, y] =z, [x, z] = [y, z] = 0 (i.e., the Heisenberg algebra). The Arens–Michael envelope U(g) ofU(g) consists of all power series

a =∑i,j,k

aijkyizjxk,

such that for any ρ > 0,

‖a‖ρ =∑i,j,k

|aijk|i!

(i + j)!ρi+j+k < ∞.

The topology on U(g) is defined by the prenorms ‖ · ‖ρ : ρ > 0.



5.2. Tensor algebras. Suppose R is an algebra and M is an R-bimodule. Using Propo-sition 4.8, it is not difficult to describe the Arens–Michael envelope of the tensor algebraTR(M). For this, we first notice that the homomorphism R

ιR−→ Ri0−→ TR(M) and the R-

bimodule morphism MιM−−→ M

i1−→ TR(M) uniquely determine (by the universal propertyof TR(M)) an R-homomorphism ιT : TR(M) → TR(M).

Proposition 5.5. Let R be an algebra and M an R-bimodule. Then the pair (TR(M), ιT )is the Arens–Michael envelope of TR(M).

Proof. Suppose ϕ : TR(M) → A is a homomorphism with values in an Arens–Michaelalgebra A. The homomorphism θ = ϕi0 : R → A extends to a homomorphism ηA =θ : R → A, which makes A an R-⊗-algebra and, in particular, an R-⊗-bimodule. There-fore the R-bimodule morphism α = ϕi1 : M → A extends to an R-bimodule morphismα : M → A, which, by Proposition 4.8, yields an R-homomorphism ϕ : TR(M) → A coin-ciding with α on M . It is now easy to see that ϕιT = ϕ, because both R-homomorphismscoincide with α on M. Thus ϕ is a homomorphism extending ϕ in the sense of Defini-tion 3.1. Its uniqueness is obvious because the image of ιT is dense in TR(M). 5.3. Ore extensions: the case δ = 0. Quantum affine spaces. As a consequence ofProposition 5.5, we can easily describe the Arens–Michael envelope of the algebra R[t; α],where R is an algebra and α is an endomorphism of it. Notice that the definition of theArens–Michael envelope implies that there exists a unique continuous endomorphism α

of R such that αιR = ιRα. Moreover, we have homomorphisms

R[t; α] → R[t; α] → Rt; α,where the former coincides with ιR on R and sends t to t, and the latter is the canonicalembedding (see Proposition 4.9). The composition of these homomorphisms will bedenoted ιR[t;α].

Corollary 5.6. Suppose R is an algebra and α is an endomorphism of it. Then the pair(Rt; α, ιR[t;α]) is the Arens–Michael envelope of R[t; α].

Proof. We shall show that the pair (Rα, ιR) is the Arens–Michael envelope of the bimod-ule R in the sense of Definition 3.2 (see also Remark 3.8). Indeed, that ιR : Rα → Rα isan R-bimodule morphism easily follows from the relation ιRα = αιR. Suppose X is anR-⊗-bimodule and ϕ : Rα → X a morphism of R-bimodules. Set x = ϕ(1); then, clearly,x · r = α(r) · x for any r ∈ R, and therefore x · r = α(r) · x for any r ∈ R. This, in turn,implies that ϕ : Rα → X, ϕ(r) = r · x is a morphism of R-bimodules extending ϕ. Theuniqueness of such extension is clear because the image of Im ιR is dense in R. Thus,Rα = Rα as R-⊗-bimodules. It remains to combine this fact with Propositions 5.5 and4.9. Example 5.3. Set R = C[x] and define an endomorphism α : R → R by the formulaα(x) = x−1. Then it is easy to see that the algebra R[y; α] is isomorphic to the universalenveloping algebra of U(g), where g is the two-dimensional Lie algebra with basis x, yand commutator [x, y] = y (see Example 5.1). By Corollary 5.6, we have an isomor-phism of topological algebras U(g) ∼= Ry; α. On the other hand, R = O(C), and itis not difficult to see that the endomorphism α : R → R acts by the formula α(f)(z) =f(z−1). Therefore, by Example 4.3, the underlying LCS of U(g) = Ry; α is isomorphicto O(C)[[y]]; under this isomorphism the coordinate function z → z from O(C) corre-sponds to x ∈ g. Thus we have a second proof of Proposition 5.2, which is independentof the results of 5.1.



Example 5.4. Fix a number q ∈ C, q = 0. Recall (see [40]) that the quantum plane isan algebra with two generators x, y, subject to the relation xy = qyx. This algebra willbe denoted Oalg

q (C2).5

More generally, let q = (qij)ni,j=1 be a complex (n × n)-matrix such that qii = 1 and

qij = q−1ji for all i, j (such matrices are usually called multiplicatively antisymmetric). A

multiparameter quantum affine space (see, for example, [1, 20]) is an algebra Oalgq (Cn)

with generators x1, . . . , xn and relations xixj = qijxjxi (i, j = 1, . . . , n). In the one-parameter case, i.e., when qij = q for all i < j, this algebra is denoted Oalg

q (Cn).It is easy to see that Oalg

q (Cn) is an iterated Ore extension

C[x1][x2; α2] · · · [xn; αn],

where for each i = 2, . . . , n, the endomorphism αi is uniquely determined by the conditionαi(xj) = qijxj (j < i). Therefore to describe the Arens–Michael envelope of Oalg

q (Cn) wecould use Corollary 5.6. However we prefer a different approach, reflecting the specificfeatures of the given algebra and leading to a more transparent description of its Arens–Michael envelope, which is not based on the “nonconstructive” formula (4.21).

First, we recall a convenient formula for the multiplication on Oalgq (Cn). For each

α ∈ Zn+, α = (α1, . . . , αn), set |α| = α1 + · · · + αn and xα = xα1

1 · · ·xαnn . Define a

function c : Zn+ × Zn

+ → C× by setting

(5.2) c(α, β) =∏i>j

qαiβj

ij .

Then in Oalgq (Cn) we have

(5.3) xαxβ = c(α, β)xα+β

(see, for example, [1, 21]).Let w : Zn

+ → R+ be an arbitrary function. For each ρ > 0 define a prenorm ‖ · ‖wρ

on Oalgq (Cn) by the formula

(5.4) ‖a‖wρ =∑α

|cα|w(α)ρ|α| for a =∑α

cαxα.

Lemma 5.7. The prenorm ‖ · ‖wρ is submultiplicative if and only if

(5.5) |c(α, β)|w(α + β) ≤ w(α)w(β) (α, β ∈ Zn+).

Proof. This follows easily from relations (5.3).

Consider the free algebra Fn with generators ξ1, . . . , ξn. Obviously, there exists aunique homomorphism Fn → Oalg

q (Cn) such that ξi → xi for each −1, . . . , n. The imageof an arbitrary element u ∈ Fn under the action of this homomorphism will be denotedu.

Fix an N ∈ Z+ and let XN ⊂ Fn denote the set of all words in ξ1, . . . , ξn of length N .The symmetric group SN acts on XN in an obvious way. Moreover, it follows from thedefinition of Oalg

q (Cn) that for any u ∈ XN and σ ∈ SN there is a unique λ(σ, u) ∈ C×

such that u = λ(σ, u)σ(u).

Lemma 5.8. The function λ : SN × XN → C× satisfies the relation

λ(στ, u) = λ(σ, τ (u))λ(τ, u) (σ, τ ∈ SN , u ∈ XN ).

5In the literature this algebra is usually denoted Oq(C2) or simply Aq; we have added the superscript

“alg” in order to use the notation Oq(C2) for an analytic analog of this algebra.



Proof. It follows from the definition of λ that

u = λ(τ, u)τ (u) and τ (u) = λ(σ, τ (u))στ (u),

whence

(5.6) u = λ(τ, u)λ(σ, τ (u))στ (u).

On the other hand,

(5.7) u = λ(στ, u)στ (u).

It remains to combine (5.6) and (5.7). Fix an α ∈ Zn

+ and consider the element ξα = ξα11 · · · ξαn

n ∈ XN, where N = |α|. Set

wq(α) = min|λ(σ, ξα)| : σ ∈ SN

.

Since λ(e, u) = 1, where e is the identity permutation, we have wq(α) ≤ 1 for any α ∈ Z+n .

Lemma 5.9. The function wq : Zn+ → R+ satisfies (5.5).

Proof. Set N1 = |α|, N2 = |β|, and N = N1 + N2. Clearly, there is a τ ∈ SN suchthat τ (ξαξβ) = ξα+β. Moreover, xαxβ = λ(τ, ξαξβ)xα+β , whence λ(τ, ξαξβ) = c(α, β).Therefore, in view of Lemma 5.8,

min|λ(σ, ξαξβ)| : σ ∈ SN

= min

|λ(στ, ξαξβ)| : σ ∈ SN

= min

|λ(σ, ξα+β)| · |λ(τ, ξαξβ)| : σ ∈ SN

= wq(α + β)|c(α, β)|.

(5.8)

Suppose now that σ1 ∈ SN1 and σ2 ∈ SN2 are such that

|λ(σ1, ξα)| = wq(α) and |λ(σ2, ξ

β)| = wq(β).

Let σ1 × σ2 ∈ SN denote the permutation

(σ1 × σ2)(i) =

σ1(i), i ≤ N1,

N1 + σ2(i − N1), i > N1.

In other words, σ1×σ2 acts on the first N1 elements as σ1 and on the remaining elementsas σ2. Clearly, (σ1 × σ2)(ξαξβ) = σ1(ξα)σ2(ξβ), and therefore

xαxβ = λ(σ1, ξα)σ1(ξα)λ(σ2, ξ

β)σ2(ξβ) = λ(σ1, ξα)λ(σ2, ξ

β)(σ1 × σ2)(ξαξβ).

On the other hand,

xαxβ = λ(σ1 × σ2, ξαξβ)(σ1 × σ2)(ξαξβ),

whence λ(σ1 × σ2, ξαξβ) = λ(σ1, ξ

α)λ(σ2, ξβ). Taking account of (5.8) we see that

wq(α)wq(β) =∣∣λ(σ1 × σ2, ξ

αξβ)∣∣

≥ min|λ(σ, ξαξβ)| : σ ∈ SN

= wq(α + β)|c(α, β)|,

Henceforth the prenorm ‖ · ‖wqρ will be denoted ‖ · ‖qρ .

Lemma 5.10. The prenorm ‖ · ‖qρ is submultiplicative and is the largest of all the sub-multiplicative prenorms ‖ · ‖ on Oalg

q (Cn) such that ‖xi‖ ≤ ρ for all i = 1, . . . , n.

Proof. First, ‖ · ‖qρ is submultiplicative by Lemmas 5.9 and 5.7. It is also clear that‖xi‖qρ = ρ for all i = 1, . . . , n. Suppose ‖ · ‖ is some other prenorm on Oalg

q (Cn) satisfyingthe assumptions of the lemma. Clearly, ‖u‖ ≤ ρN for any u ∈ XN and any N ∈ Z+.Take an arbitrary α ∈ Zn

+, set N = |α|, and choose σ ∈ SN such that wq(α) = |λ(σ, ξα)|.Since xα = λ(σ, ξα)σ(ξα), we have

‖xα‖ = wq(α)‖σ(ξα)‖ ≤ wq(α)ρ|α|.



This, together with the explicit description of ‖ · ‖qρ , implies that ‖a‖ ≤ ‖a‖qρ for anya ∈ Oalg

q (Cn).

Suppose, as before, that q = (qij) is a multiplicatively antisymmetric complex(n × n)-matrix. Set

Oq(Cn) =

a =∑

α∈Zn+

cαxα : ‖a‖qρ =∑

α∈Zn+

|cα|wq(α)ρ|α| < ∞ ∀ρ > 0

.

If qij = q for all i < j, then the corresponding space will be denoted Oq(Cn), the functionwq will be denoted wq, and the prenorm ‖ · ‖qρ will be denoted ‖ · ‖q

ρ.Clearly, Oq(Cn) is a Frechet space with respect to the topology defined by the system of

prenorms ‖·‖qρ : ρ > 0. Notice that assigning to each a =∑

α cαxα ∈ Oq(Cn) the func-tion f : Cn → C, f(z) =

∑α cαwq(α)zα we have an LCS isomorphism between Oq(Cn)

and the space of entire functions O(Cn).

Theorem 5.11. There is a unique multiplication on Oq(Cn) making it a topologicalalgebra and such that xixj = qijxjxi for all i, j = 1, . . . , n. Algebra thus obtained,together with the tautological embedding of Oalg

q (Cn) in it, is the Arens–Michael envelopeof Oalg

q (Cn). Moreover, each prenorm of the system ‖ · ‖qρ : ρ > 0 defining the topologyon Oq(Cn) is submultiplicative.

Proof. By Lemma 5.10, the prenorm ‖·‖qρ is submultiplicative on Oalgq (Cn). Since Oq(Cn)

is, obviously, a completion of Oalgq (Cn) with respect to the system ‖ · ‖qρ : ρ > 0, we

see that the multiplication on Oalgq (Cn) extends uniquely to a continuous multiplication

on Oq(Cn) and makes the latter an Arens–Michael algebra.Suppose now that A is an Arens–Michael algebra and ϕ : Oalg

q (Cn) → A is a ho-momorphism. Fix a continuous submultiplicative prenorm ‖ · ‖ on A and set ρ =max1≤i≤n ‖ϕ(xi)‖. Applying Lemma 5.10, we have ‖ϕ(a)‖ ≤ ‖a‖qρ for any a ∈ Oalg

q (Cn).Therefore ϕ is continuous in the topology defined by the system ‖ · ‖qρ : ρ > 0. Therest is clear.

We shall now examine the two simplest cases when wq can be computed explicitly.

Proposition 5.12. (i) If |qij | ≥ 1 for all i < j, then wq(α) = 1 for all α ∈ Zn+.

(ii) If |qij | ≤ 1 for all i < j, then wq(α) = |c(α, α)|−1 for all α ∈ Zn+.

Proof. (i) It follows from the assumption and (5.2) that |c(α, β)| ≤ 1 for all α, β ∈ Zn+.

Therefore the function ε : Zn+ → Z+, ε(α) ≡ 1, satisfies the condition of Lemma 5.7.

Therefore, ‖ · ‖ε1 is submultiplicative on Oalg

q (Cn). Since wq(α) ≤ 1 for any α ∈ Zn+, we

have ‖ · ‖q1 ≤ ‖ ·‖ε1. On the other hand, Lemma 5.10 yields the opposite inequality. Thus,

‖ · ‖q1 = ‖ · ‖ε1, which, in turn, is equivalent to the equality wq(α) = 1 for all α ∈ Zn

+.(ii) As before, consider the free algebra Fn with generators ξ1, . . . , ξn. For any i =

1, . . . , n set ηi = ξn−i. Define also a multiplicatively antisymmetric (n × n)-matrixq = (qij) by setting qij = qn−i,n−j . Clearly, for any α ∈ Zn

+ with |α| = N we have

(5.9) min|λ(σ, ηα)| : σ ∈ SN

= wq(α).

Define a permutation τ ∈ SN by the formula τ (i) = N − i (i = 1, . . . , N). Clearly,τ (ξα) = ηα, where α = (αn, . . . , α1). Furthermore, the multiplication law on Oalg

q (Cn)shows that

(5.10) xα11 · · ·xαn

n =∏i<j

qαiαj

ij xαnn · · ·xα1

1 = c(α, α)−1xαnn · · ·xα1

1 ,



or, equivalently, λ(τ, ξα) = c(α, α)−1. Therefore, in view of Lemma 5.8 and equality(5.9), we have

wq(α) = min|λ(σ, ξα)| : σ ∈ SN

= min

|λ(στ, ξα)| : σ ∈ SN

= min

|λ(σ, ηα)| : σ ∈ SN

· |λ(τ, ξα)| = wq(α)|c(α, α)|−1.

But wq ≡ 1 by (i) and therefore wq(α) = |c(α, α)|−1.

Corollary 5.13. (i) If |qij | ≥ 1 for i < j, then

(5.11) Oq(Cn) =

a =∑

α∈Zn+


α∈Zn+

|cα|ρ|α| < ∞ ∀ρ > 0

.

(ii) If |qij | ≤ 1 for i < j, then

Oq(Cn) =

a =∑

α∈Zn+


α∈Zn+

|cα| · |c(α, α)|−1ρ|α| < ∞ ∀ρ > 0

.

In both cases the topology on Oq(Cn) is defined by the system of submultiplicativeprenorms ‖ · ‖qρ : ρ > 0 and the multiplication is uniquely determined by the relations

xixj = qijxjxi (i, j = 1, . . . , n).

Remark 5.2. Notice that assertion (i) of the preceding corollary can be deduced byinduction from Corollary 5.6, Proposition 4.11, and Proposition 4.6.

In the case of the quantum plane, Corollary 5.13 becomes especially simple.

Corollary 5.14. Suppose q ∈ C \ 0.(i) If |q| ≥ 1, then

Oq(C2) =

a =∞∑

i,j=0

cijxiyj : ‖a‖q

ρ =∞∑

i,j=0

|cij |ρi+j < ∞ ∀ρ > 0

.

(ii) If |q| ≤ 1, then

Oq(C2) =

a =∞∑

i,j=0

cijxiyj : ‖a‖q

ρ =∞∑

i,j=0

|cij ||q|ijρi+j < ∞ ∀ρ > 0

.

In both cases the topology on Oq(C2) is defined by the system of submultiplicative prenorms‖ · ‖q

ρ : ρ > 0 and the multiplication is uniquely determined by the relation xy = qyx.

5.4. Connections with the algebra of holomorphic functions on the quantumball. Let Bρ ⊂ Cn be the closed ball in Cn of radius ρ > 0 and center at the origin. LetA(Bρ) denote the Banach algebra consisting of all continuous functions on Bρ which areholomorphic on its interior. Fix an arbitrary R > 1 and consider the inverse system ofBanach algebras

(5.12) A(B1) ← A(BR) ← A(BR2) ← A(BR3) ← · · · ,

where the arrows denote the restriction homomorphisms. It is easy to see that the inverselimit of (5.12) is, up to isomorphism of topological algebras, the algebra O(Cn) of entirefunctions.

Let z1, . . . , zn be the coordinates on Cn. Notice that (5.12) is isomorphic to the inversesystem

(5.13) A(B)γr←− A(B)

γr←− A(B)γr←− A(B)

γr←− · · · ,



where B = B1, r = R−1, and γr denotes the unique continuous endomorphism of A(B)such that γr(zi) = rzi for all i = 1, . . . , n. Thus

(5.14) O(Cn) ∼= lim←−(A(B), γr

).

Below we shall show that (5.14) has a quantum analog, in which O(Cn) is replacedby Oq(Cn) (see above) and the role of the algebra A(B) is played by the algebra ofholomorphic functions on the quantum ball, introduced by Vaksman [70].

First, we recall some definitions from [55] and [70]. Fix q ∈ (0, 1) and let Aq denotethe algebra with generators a1, . . . , an, a+

1 , . . . , a+n and relations

a+i a+

j = qa+j a+

i (i < j),

aja+i = qa+

i aj (i = j),

ajai = qaiaj (i < j),

aia+i = 1 + q2a+

i ai − (1 − q2)∑k>i

a+k ak.

This algebra was introduced by W. Pusz and S. Woronowicz [55]; this is a q-analog ofthe algebra of canonical commutation relations. There is a unique involution a → a∗

on Aq such that a∗i = a+

i for all i = 1, . . . , n.Notice that Aq is a particular case of the quantum Weyl algebra (see [14, 29]; in the

simplest case of n = 1 the quantum Weyl algebra is discussed below in Example 5.5). Thisalgebra can be viewed as an iterated Ore extension of the base field C; as a consequence,the ordered monomials in a±

1 , . . . , a±n form a basis of Aq (ibid.). This, in turn, implies

that a+1 , . . . , a+

n generate a subalgebra of Aq isomorphic to Oalgq (Cn).

Let H be a Hilbert space with orthonormal basis eα : α ∈ Zn+. For each α =

(α1, . . . , αn) ∈ Zn+ the basis element eα will also be denoted |α1, . . . , αn〉. For any k ∈ Z+

we set

[k] =1 − q2k

1 − q2, [k]! = [1] · [2] · · · [k].

Finally, for α = (α1, . . . , αn) ∈ Zn+ set

[α]! = [α1]! · · · [αn]! .

By [55], there is an irreducible ∗-representation π : Aq → B(H), uniquely determinedby the formulas

π(aj)|α1, . . . , αn〉 =√

[αj ] q∑

k>j αk |α1, . . . , αj − 1, . . . , αn〉,(5.15)

π(a+j )|α1, . . . , αn〉 =

√[αj + 1] q

∑k>j αk |α1, . . . , αj + 1, . . . , αn〉.(5.16)

This is a q-analog of the classical Fock representation. Notice that, unlike the classicalcase, all the operators π(a) with a ∈ Aq are bounded and ‖π(a±

j )‖ ≤ (1− q2)−1/2 for allj = 1, . . . , n. (It is easy to check that the last inequality is in fact an equality, but wewill not need this fact.)

D. P. Proskurin and Yu. S. Samoılenko [52] showed that π is faithful and that theC∗-algebra π(Aq) ⊂ B(H) is the universal enveloping C∗-algebra of Aq in the sense thatany ∗-homomorphism from Aq to a C∗-algebra uniquely factors through π(Aq). As wasnoticed by L. L. Vaksman [70], the algebra C(B)q = π(Aq) can be viewed as a q-analogof the algebra C(B) of continuous functions on the closed ball B, whereas its closedsubalgebra A(B)q generated by the operators π(a+

k ) (k = 1, . . . , n) can be viewed as aq-analog of the algebra A(B).



Henceforth we shall use the notation introduced in the preceding section. Recall, inparticular, that for any α ∈ Z+ we have

(5.17) wq(α) = |c(α, α)|−1 = q∑

i<j αiαj

(see Proposition 5.12 (ii)).Define a homomorphism

ϕ0 : Oalgq (Cn) → A(B)q, xi →

√1 − q2 π(a+

i ) (i = 1, . . . , n).

As follows from the above remarks, it is injective. Henceforth we shall identify Oalgq (Cn)

with a dense subalgebra of A(B)q via ϕ0. The corresponding norm on Oalgq (Cn) will be

denoted ‖ · ‖; the norm ‖ · ‖qρ on Oalg

q (Cn), introduced in the previous section for anyρ > 0 (see Corollary 5.13 (ii)) will be denoted ‖ · ‖ρ.

Lemma 5.15. (i) ‖a‖ ≤ ‖a‖1 for any a ∈ Oalgq (Cn).

(ii) For any r <√

1 − q2 there is a Cr > 0 such that ‖a‖r ≤ Cr‖a‖ for any a ∈Oalg

q (Cn).

Proof. Since ‖π(a+i )‖ ≤ (1 − q2)−1/2 (see above), we have ‖xi‖ ≤ 1 for any i = 1, . . . , n.

In view of Lemma 5.10, this yields (i).To prove (ii), notice that by induction one easily deduces from (5.16) that

π(a+1 )α1 · · ·π(a+

n )αne0 =√

[α]! q∑

i<j αiαj eα =√

[α]! wq(α)eα.

Therefore,xαe0 =

√[α]! (1 − q2)|α|/2wq(α)eα,

and for any a =∑

α cαxα ∈ Oalgq (Cn) we have an estimate

‖a‖2 ≥ ‖ae0‖2 =∑α

|cα|2(1 − q2)|α|w2q(α)[α]! ≥ sup

α|cα|2(1 − q2)|α|w2

q(α).

It now follows that

‖a‖r =∑α

|cα|wq(α)r|α| =∑α

( r√1 − q2

)|α||cα|(1 − q2)|α|/2wq(α) ≤ Cr‖a‖,

where

Cr =∑α

( r√1 − q2

)|α|=( 1

1 − r/√

1 − q2

)n.

This proves (ii).

Theorem 5.16. For any r <√

1 − q2 there exists a unique continuous homomorphismγr : A(B)q → A(B)q such that γr(xi) = rxi for all i = 1, . . . , n. The inverse limit of thesystem of Banach algebras

(5.18) A(B)qγr←− A(B)q

γr←− A(B)qγr←− A(B)q

γr←− · · ·is isomorphic to Oq(Cn).

Proof. For any s > 0 consider a homomorphism

γs : Oalgq (Cn) → Oalg

q (Cn), xi → sxi (i = 1, . . . , n).

It is easy to see that

(5.19) ‖γs(a)‖t = ‖a‖st

(a ∈ Oalg

q (Cn), s, t > 0).

Fix an arbitrary r <√

1 − q2. It follows from Lemma 5.15 and (5.19) that

‖γr(a)‖ ≤ ‖γr(a)‖1 = ‖a‖r ≤ Cr‖a‖



for any a ∈ Oalgq (Cn). This and the fact that Oalg

q (Cn) is dense in A(B)q imply theexistence and uniqueness of the homomorphism γr with the required properties.

For the sake of convenience, label the terms of (5.18) by nonnegative integers. Moreprecisely, for any k ∈ Z+ set Ak = A(B)q and define homomorphisms τk+1

k : Ak+1 → Ak

by setting τk+1k = γr for all k ∈ Z+. Set A = lim←−(Ak, τk+1

k ), and let τk denote thecanonical morphism from A to Ak. Define a prenorm ‖·‖k on A by setting ‖a‖k = ‖τk(a)‖for any a ∈ A. Notice that the sequence of prenorms ‖·‖k : k ∈ Z+ defines the standardinverse limit topology on A.

Set R = r−1 and for any k ∈ Z+ define a homomorphism

ϕk : Oalgq (Cn) → Ak, xi → Rkxi (i = 1, . . . , n).

Clearly, ϕk = τk+1k ϕk+1 for any k ∈ Z+. Therefore, there exists a unique homomorphism

ϕ : Oalgq (Cn) → A such that ϕk = τkϕ for all k ∈ Z+. In turn, it gives rise to a

homomorphism ϕ : Oq(Cn) → A.We shall show that ϕ is an isomorphism. For this, we first notice that for any k ∈ Z+

the image of ϕk is dense; hence the same is true for ϕ. Therefore, to prove that ϕ is anisomorphism, it suffices to check that ϕ is topologically injective.

Using (5.19) and Lemma 5.15, we have

‖a‖Rk = ‖γRk+1(a)‖r ≤ Cr‖γRk+1(a)‖ = Cr‖ϕk+1(a)‖ = Cr‖τk+1ϕ(a)‖ = Cr‖ϕ(a)‖k+1

for any k ∈ Z+ and any a ∈ Oalgq (Cn). Since the topology on Oalg

q (Cn) inducedfrom Oq(Cn) is defined by ‖ · ‖Rk : k ∈ Z+, it follows from the last computationthat ϕ is topologically injective. As we mentioned before, this implies that ϕ is anisomorphism.

5.5. Ore extensions: the “general” case. Quantum Weyl algebras and quan-tum matrices. In the general case, i.e., when α = 1R and δ = 0, there does not seem tobe a satisfactory description of the Arens–Michael envelope of the Ore extension R[t; α, δ]as an algebra of “skew power series” (cf. Example 3.4). Below we shall describe a specialbut often encountered situation when such a description is in fact possible and we shallalso illustrate it with several examples.

Suppose R is an algebra, α : R → R is an endomorphism, and δ : R → R is an α-derivation. As before (see Section 5.3), α will denote the (unique) endomorphism of Rsuch that

(5.20) αιR = ιRα.

Clearly, ιR can be viewed as an R-bimodule morphism αR → αR. Therefore, ιRδ : R →αR is a derivation. It follows from Corollary 3.9 that there is a unique derivation δ : R →αR, i.e., a unique α-derivation of R such that

(5.21) διR = ιRδ.

In the next theorem, as in Section 4.1, z ∈ O(C, R) denotes the function sending λ ∈ C

to λ1R ∈ R.

Theorem 5.17. Suppose that the family α, δ is m-localizable. Then there exists aunique R-homomorphism

(5.22) ιR[t;α,δ] : R[t; α, δ] → O(C, R; α, δ) such that t → z.

The algebra O(C, R; α, δ) with the homomorphism (5.22) is the Arens–Michael envelopeof R[t; α, δ].



Proof. The existence and uniqueness of the R-homomorphism (5.22) directly followsfrom the universal property of Ore extensions and relations (5.20) and (5.21). Further-more, the algebra O(C, R; α, δ) is Arens–Michael by Proposition 4.5. Suppose now thatϕ : R[t; α, δ] → A is a homomorphism to some Arens–Michael algebra A. Set ψ = ϕ|Rand x = ϕ(t). The homomorphism ψ : R → A extending ψ makes A an R-⊗-algebra;moreover, since ϕ is a homomorphism, it follows from (5.20) and (5.21) that

x · r = α(r) · x + ψ(δ(r))

for any r ∈ Im ιR. Since Im ιR is dense in R, the latter equality holds for any r ∈R. Therefore, by Proposition 4.4 (see also Remark 4.6), there exists a unique R-homomorphism (or, equivalently, R-homomorphism) ϕ : O(C, R; α, δ) → A such thatϕ(z) = x. This equality and the fact that ϕ is an R-homomorphism is equivalent to therelation ϕ ιR[t;α,δ] = ϕ. The rest is clear.

To apply the preceding theorem in concrete situations it is useful to have a purely al-gebraic condition guaranteeing that the family α, δ is m-localizable. The next theoremgives one such possible condition of that kind.

Theorem 5.18. Suppose A is an algebra with generators x1, . . . , xn+1 and relations

(5.23)

⎧⎪⎨⎪⎩xixj = qijxjxi (i, j ≤ n),

xn+1xi = λixixn+1 (2 ≤ i ≤ n),

xn+1x1 = λ1x1xn+1 + µxγ11 · · ·xγn

n ,

where (qij)ni,j=1 = q is a multiplicatively antisymmetric (n×n)-matrix, λ1, . . . , λn, µ ∈ C,

and γ = (γ1, . . . , γn) ∈ Zn+. For any j = 1, . . . , n let ej ∈ Zn

+ denote the element with 1at the j-th place and 0 elsewhere. Define a function c : Zn

+ × Zn+ → C× by (5.2) and

assume that

(5.24) λ1 = c(γ, e1), λj =c(γ, ej)

q1jc(ej , γ)(j ≥ 2).

Moreover, assume that

(5.25) |λi| ≤ 1 (i = 1, . . . , n), |qij | ≥ 1 (i < j).

Then the Arens–Michael envelope of A is isomorphic, as a locally convex space, to thespace of entire functions

O(Cn+1) =

a =∑

α∈Zn+1+

cαxα : ‖a‖ρ =∑

α∈Zn+1+

|cα|ρ|α| < ∞ ∀ρ > 0

,

and the multiplication on it is uniquely determined by (5.23).

Proof. Suppose F is the free algebra with generators ξ1, . . . , ξn, and let I ⊂ F be thetwo-sided ideal generated by the relations ξiξj = qijξjξi (i, j = 1, . . . , n). Set R = F/I,let π denote the quotient homomorphism from F to R, and set yi = π(ξi) (i = 1, . . . , n).As R = Oalg

q (Cn), the multiplication law (5.3) (with x replaced by y) holds on R.It is easy to see that there are a unique endomorphism σ and a unique σ-derivation δ

of F such that

σ(ξi) = λiξi (1 ≤ i ≤ n), δ(ξ1) = µξγ , δ(ξk) = 0 (k ≥ 2).

Clearly, σ(I) ⊂ I. We shall show that δ(I) ⊂ I. Indeed,

δ(ξiξj − qijξjξi) = 0 when i, j ≥ 2,



and

πδ(ξ1ξj − q1jξjξ1) = π(δ(ξ1)ξj − q1j σ(ξj)δ(ξ1)

)= µπ

(ξγξj − q1jλjξjξ

γ)

= µ(c(γ, ej) − q1jλjc(ej , γ)

)yej+γ

for any j ≥ 2. But the latter expression is zero by (5.24). In view of the relationqj1 = q−1

1j , this yields δ(I) ⊂ I.Thus σ and δ fix I and therefore give rise to linear maps σ, δ : R → R. Moreover, it is

easy to see that σ is an endomorphism and δ is a σ-derivation of R.Consider the Ore extension B = R[yn+1; σ, δ]. The constructions of R, σ, and δ show

that y1, . . . , yn+1 satisfy the same relations (5.23) as the xi. Therefore, there exists aunique homomorphism A → B such that xi → yi (i = 1, . . . , n + 1). On the other hand,the universal property of Ore extensions yields a unique homomorphism B → A suchthat yi → xi (i = 1, . . . , n + 1). This implies that A and B are isomorphic. Henceforthwe shall assume that A = B and xi = yi (i = 1, . . . , n + 1).

We shall show that σ and δ satisfy the conditions of Theorem 5.17. Since R = Oalgq (Cn)

and |qij | ≥ 1 for i < j, it follows from Corollary 5.13 that

R =

r =∑

β∈Zn+

cβxβ : ‖r‖ρ =∑

β∈Zn+

|cβ|ρ|β| < ∞ ∀ρ > 0

,

and all the prenorms ‖ · ‖ρ (ρ > 0) are submultiplicative. Since σ(xi) = λixi and |λi| ≤ 1for all i = 1, . . . , n, we have

(5.26) ‖σ(r)‖ρ ≤ ‖r‖ρ

for all r ∈ R and all ρ > 0.It is easily checked by induction that

δ(xm1 ) = µ

c(γ, e1)m − λm1

c(γ, e1) − λ1x(m−1)e1+γ

for all m ∈ Z+. This and the fact that δ(xk) = 0 for k > 1 imply that

(5.27) δ(xβ) = δ(xβ11 )xβ2

2 · · ·xβnn = µ

c(γ, e1)β1 − λβ11

c(γ, e1) − λ1x(β1−1)e1+γxβ2

2 · · ·xβnn

for all β ∈ Zn+, β1 > 0, and

(5.28) δ(xβ) = 0 when β1 = 0.

Furthermore, the condition |qij | ≥ 1 for i > j and (5.2) imply that |c(γ, e1)| ≤ 1. Since|λ1| ≤ 1, it follows from (5.27) and (5.28) that

‖δ(xβ)‖ρ ≤ Cρρ|β|, where Cρ =

2µρ|γ|−1

|c(γ, e1) − λ1|ρ|β|,

for all β ∈ Zn+ and all ρ > 0. This and the explicit description of ‖ · ‖ρ imply that

‖δ(r)‖ρ ≤ Cρ‖r‖ρ

for all r ∈ R and all ρ > 0. Comparing this with (5.26), we have that the family σ, δ ism-localizable. To finish the proof, apply Theorem 5.17 and the canonical isomorphismof locally convex spaces

O(C, R) ∼= O(C, O(Cn)) ∼= O(C × Cn) = O(Cn+1).



Example 5.5 (quantum Weyl algebras). Recall (see, for example, [20]) that the quantumWeyl algebra (with parameter q ∈ C \ 0) is the algebra A1(q) with generators x, ∂ andrelation ∂x − qx∂ = 1. When q = 1 we have the usual Weyl algebra (see Example 3.4).

Suppose that |q| ≤ 1 and q = 1. Then it is not difficult to see that A1(q) satisfies theconditions of Theorem 5.18; moreover, n = 1 (and therefore c ≡ 1), λ1 = q, µ = 1, andγ = 0. If |q| ≥ 1 but still q = 1, then it is convenient for our purposes to represent A1(q)as the algebra with generators ∂, x and relation x∂ − q−1∂x = −q−1. It also satisfiesthe conditions of Theorem 5.18 with n = 1, λ1 = q−1, µ = −q−1, and γ = 0. Applyingthis theorem, we have the following description of the Arens–Michael envelope of thequantum Weyl algebra.

Corollary 5.19. Suppose q ∈ C \ 0 and q = 1.(i) If |q| ≤ 1, then

A1(q) =

a =∞∑

i,j=0

cijxi∂j : ‖a‖ρ =

∞∑i,j=0

|cij |ρi+j < ∞ ∀ρ > 0

.

(ii) If |q| ≥ 1, then

A1(q) =

a =∞∑

i,j=0

cij∂ixj : ‖a‖ρ =

∞∑i,j=0

|cij |ρi+j < ∞ ∀ρ > 0

.

In both cases the topology on A1(q) is defined by the system of prenorms ‖ · ‖ρ : ρ > 0,and the multiplication is uniquely determined by the relation ∂x − qx∂ = 1.

Remark 5.3. Thus we see that for all q = 1, the Arens–Michael envelopes of the corre-sponding quantum Weyl algebras are similar. In particular, their underlying LCS areisomorphic to the space of entire functions O(C2), and the canonical morphism from A1(q)to A1(q) is injective. The case q = 1 is thus degenerate: recall that A1(1) = 0 (seeExample 3.4).

Remark 5.4. It is not difficult to check that if q is not a root of unity, then A1(q) can berealized as the algebra of q-differentiable operators on C[x] with polynomial coefficients:

A1(q) =

a =N∑

i=0

an(x)Dnq : an ∈ C[x], N ∈ Z+

,

where the q-derivative Dq is given by

(Dqf)(x) =f(qx) − f(x)

qx − x(f ∈ C[x]).

On the other hand, it is easy to check that when |q| ≤ 1, q = 1, the operator Dq canbe viewed as a bounded operator on the Hardy space H2. Therefore, if |q| ≤ 1 and isnot a root of unity, then the quantum Weyl algebra A1(q) has a faithful representationπ : A1(q) → B(H2), which is uniquely determined by the conditions

(π(x)f)(z) = zf(z), (π(∂)f)(z) = (Dqf)(z) (f ∈ H2, |z| < 1).

In particular, for the above values of q we have a simple proof that the Arens–Michaelenvelope of A1(q) is not trivial, which is not based on Theorem 5.18.

Example 5.6 (quantum matrices). Suppose q ∈ C \ 0. Recall (see, for example, [31]),that the algebra of quantum (2×2)-matrices Mq(2) has generators a, b, c, d and relations

ab = qba, ac = qca, bc = cb,

bd = qdb, cd = qdc, ad − da = (q − q−1)bc.(5.29)



Set x1 = a, x2 = b, x3 = c, and x4 = d. Then it is not difficult to see that Mq(2)belongs to the type of algebras considered in Theorem 5.18. Moreover, n = 3, λ1 = 1,λ2 = λ3 = q−1, µ = q−1 − q, γ = (0, 1, 1), and

q =

⎛⎝ 1 q qq−1 1 1q−1 1 1

⎞⎠ .

It is easy to see that

c(γ, e1) = q−2, c(γ, e2) = c(e2, γ) = c(γ, e3) = c(e3, γ) = 1,

so that conditions (5.24) hold if and only if q = ±1. It is also clear that conditions (5.25)hold if and only if |q| ≥ 1.

To cover the case |q| ≤ 1, consider the algebra Mq−1(2). To distinguish its standardgenerators from the generators a, b, c, d of Mq(2), denote them a′, b′, c′, d′, respectively.Then it is easy to see that there is an isomorphism Mq(2) → Mq−1(2) such that a → d′,d → a′, b → b′, c → c′.

Corollary 5.20. Suppose q ∈ C \ 0.(i) If |q| ≥ 1, then

Mq(2) =

u =∞∑

i, j, k, l=0

cijklaibjckdl : ‖u‖ρ =

∞∑i, j, k, l=0

|cijkl|ρi+j+k+l < ∞ ∀ρ > 0

.

(ii) If |q| ≤ 1, then

Mq(2) =

u =∞∑

i, j, k, l=0

cijkldibjckal : ‖u‖ρ =

∞∑i, j, k, l=0

|cijkl|ρi+j+k+l < ∞ ∀ρ > 0

.

In both cases the topology on Mq(2) is defined by the system of prenorms ‖ ·‖ρ : ρ > 0,and the multiplication is uniquely determined by relations (5.29).

Proof. In view of the above, in the case q = ±1 the proof reduces to applying Theorem5.18. If q = 1, then Mq(2) = C[a, b, c, d], and there is nothing to prove (see Example 3.1).If q = −1, then it is not difficult to see that Mq(2) = O

algq′ (C4), where

(5.30) q′ =

⎛⎜⎜⎝1 −1 −1 1

−1 1 1 −1−1 1 1 −1

1 −1 −1 1

⎞⎟⎟⎠ .

Thus in the case q = −1 the proof reduces to applying Corollary 5.13.

5.6. Skew Laurent extensions. Quantum tori. Let R be an algebra, α : R → R

an automorphism, and α its canonical extension to R (see (5.20)). Clearly, α is anautomorphism of R and α−1 = (α−1) .

Theorem 5.21. Suppose that the family α, α−1 is m-localizable. Then there exists aunique R-homomorphism

(5.31) ιR[t,t−1;α] : R[t, t−1; α] → O(C×, R; α) such that t → z.

The algebra O(C×, R; α) with the homomorphism (5.31) is the Arens–Michael envelopeof R[t, t−1; α].

In view of Propositions 4.15 and 4.14, the proof of this theorem is similar to the proofof Theorem 5.17.



Example 5.7. Suppose q = (qij)ni,j=1 is a multiplicatively antisymmetric (n×n)-matrix.

Recall that the multiparameter quantum torus (see, for example, [41, 20]) is the alge-bra Oalg

q

((C×)n

)with generators x1, x

−11 , . . . , xn, x−1

n and relations xixj = qijxjxi andxix

−1i = x−1

i xi = 1 (i, j = 1, . . . , n).

Corollary 5.22. Let q = (qij)ni,j=1 be a multiplicatively antisymmetric (n × n)-matrix

such that |qij | = 1 for all i, j. Let x1, . . . , xn denote the coordinates on Cn. ThenO((C×)n

)admits a unique multiplication making it a ⊗-algebra such that xixj = qijxjxi

and xix−1i = x−1

i xi = 1 for all i, j = 1, . . . , n. The algebra thus obtained, togetherwith the tautological embedding of Oalg

q

((C×)n

)in it, is the Arens–Michael envelope of

Oalgq

((C×)n

). Its topology is defined by the family of submultiplicative prenorms

(5.32) ‖a‖ρ =∑

α∈Zn

|cα|ρ|α| (ρ > 0),

where a =∑

α cαxα is the Laurent expansion of a ∈ O((C×)n

).

Proof. We induct on n. When n = 0, the assertion is obvious. Suppose that it is truefor matrices of order n− 1. Set q = (qij)n−1

i,j=1 and R = Oalgq

((C×)n−1

). It is not difficult

to see that Oalgq

((C×)n

)= R[xn, x−1

n ; α], where the automorphism α : R → R is uniquelydetermined by the conditions α(xi) = qnixi (i = 1, . . . , n−1). The induction assumptionand the condition |qni| = 1 imply that ‖α(a)‖ρ = ‖a‖ρ for any a ∈ R and any ρ > 0.It remains to apply Theorem 5.21, Proposition 4.16 and the canonical isomorphism oflocally convex spaces

O(C×, R) ∼= O(C×, O

((C×)n−1

)) ∼= O(C× × (C×)n−1

)= O((C×)n

).

Definition 5.1. The Arens–Michael envelope of Oalgq

((C×)n

)described in the preceding

proposition will be denoted Oq

((C×)n

).

When the condition |qij | = 1 fails for at least one pair of indices i, j, the Arens–Michaelenvelope of the quantum torus degenerates:

Proposition 5.23. Suppose that |qij | = 1 for some i, j. Then Oalgq

((C×)n

)= 0.

Proof. It suffices to show that there are no homomorphisms from Oalgq

((C×)n

)to nonzero

Banach algebras. Suppose that ϕ : Oalgq

((C×)n

)→ A is such a homomorphism. Set

u = ϕ(xi), v = ϕ(xj), and q = qij . The spectrum of an arbitrary element a ∈ A willbe denoted σ(a). We have that u, v ∈ A are invertible and uv = qvu. Therefore, settingK = σ(uv) and using the fact that σ(uv) = σ(vu) (see, for example, [24, 2.1.8]), wehave that K = qK. Since K is a nonempty compact and |q| = 1, the last relation isonly possible when K = 0. But this is also impossible since uv is invertible. Thecontradiction obtained shows that A = 0.

6. Homological epimorphisms

Let A, B be categories and F : A → B a covariant functor. Recall that F is said tobe fully faithful if it is full and faithful (see [39] for terminology), i.e., if it gives rise toa bijection between Hom A(X, Y ) and Hom B(F (X), F (Y )) for any X, Y ∈ Ob (A). Inother words, F is fully faithful if and only if it gives rise to an equivalence between A

and a full subcategory of B.

Proposition 6.1. Let f : A → B be a homomorphism of ⊗-algebras. The followingconditions are equivalent:

(i) f is an epimorphism in the category of ⊗-algebras;



(ii) the functor f• : B-mod → A-mod is fully faithful;(iii) the canonical morphism B ⊗A B → B is an isomorphism of LCS;(iv) for any X ∈ Ob (mod-B) and any Y ∈ Ob (B-mod) the canonical morphism

X ⊗A Y → X ⊗B Y is an isomorphism of LCS.

Proof. The proof that (i)⇐⇒(ii)⇐⇒(iii) is a verbatim repetition of the proof given in [64,XI.1] for the category of rings. The implication (iv)=⇒(iii) is obvious, and (iv) can beobtained from (iii) by tensoring B with X on the left and with Y on the right.

Remark 6.1. In [67], ⊗-algebra epimorphisms are called pseudo-quotient homomorphisms.

Lemma 6.2. Suppose A, B are exact categories and F : A → B is an exact additivecovariant functor. The following conditions are equivalent:

(i) RF : Db(A) → Db(B) is fully faithful;(ii) for any X, Y ∈ Ob (A) and any n ∈ Z+ the canonical morphism Ext n

A(X, Y ) →Ext n

B(F (X), F (Y )) is a bijection.

The proof of this lemma given in [17] for abelian categories carries over verbatim toexact categories.

Theorem 6.3. Suppose f : A → B is an R-S-homomorphism from an R-⊗-algebra A toan S-⊗-algebra B. The following conditions are equivalent:

(1) Rf• : Db((B, S)-mod) → Db((A, R)-mod) is fully faithful;(2) for any X, Y ∈ Ob ((B, S)-mod) and any n ∈ Z+ the canonical morphism

Ext nB,S(X, Y ) → Ext n

A,R(X, Y ) is bijective;(3) for any X ∈ Ob (Db((B, S)-mod)) the canonical morphism

B ⊗LA(Rf•(X)) → X

is an isomorphism in Db((B, S)-mod);(4) f is an epimorphism of ⊗-algebras and, moreover, any X ∈ Ob ((B, S)-mod),

viewed as an object of (A, R)-mod, is acyclic relative to the functor

B ⊗A ( · ) : (A, R)-mod → (B, S)-mod .

Proof. (1)⇐⇒(2): This is a particular case of Lemma 6.2.(1)⇐⇒(3): Since B ⊗A ( · ) is left adjoint to f• : (B, S)-mod → (A, R)-mod, the same

is true for their derived functors [32, Lemma 15.6]. It remains to apply Theorem 4.3.1from [39].

(3)=⇒(4): Taking into account that f• is exact, it follows from (3) that for anyX ∈ Ob ((B, S)-mod), the canonical morphism B ⊗L

A X → X is an isomorphism inDb((B, S)-mod) and therefore in Db(LCS). Applying to it the functor H0, we have thatthe composition of the canonical morphisms

Tor A,R0 (B, X) → B ⊗A X → X

is an isomorphism in LCS. Since the first morphism in this composition is the canonicalmap from an LCS to its completion (see Section 2), we conclude that both morphismsare isomorphisms in LCS. Therefore, B ⊗A X → X is an isomorphism in B-mod. Therest is clear.

(4)=⇒(2): Suppose P• is a projective resolution of X in (A, R)-mod. By (4), B ⊗A P•is a projective resolution of X in (B, S)-mod. Therefore

Ext nA,R(X, Y ) ∼= Hn(Ah(P•, Y )) ∼= Hn

(Bh(B ⊗A P•, Y )

) ∼= Ext nB,S(X, Y ).



Definition 6.1. If the homomorphism f : A → B satisfies the conditions of the precedingtheorem, then we shall call it a left relative homological epimorphism. The homomorphismf : A → B is called a right relative homological epimorphism if f : Aop → Bop is a leftrelative homological epimorphism.

Lemma 6.4. Suppose f : A → B is an R-S-homomorphism from an R-⊗-algebra A toan S-⊗-algebra B which is an epimorphism of ⊗-algebras. Let A denote any of the exactcategories

(A, R)-mod-(A, R), (A, A)-mod-(A, R), (A, R)-mod-(A, A),and B any of the exact categories

(B, S)-mod-(B, S), (B, B)-mod-(B, S), (B, S)-mod-(B, B).

Consider the functor

(6.1) T = B ⊗A ( · ) ⊗A B : A → B.

Then, if A is T -acyclic for some choice of the exact categories Aand B, then it is alsoT -acyclic for any other choice of those categories.

Proof. Suppose that (2.5) is a projective resolution of A in (A, R)-mod-(A, R). In viewof Proposition 2.6, it is also a projective resolution of A in (A, R)-mod-(A, A) and in(A, A)-mod-(A, R). Since f is an epimorphism, it is easy to see that T (A) ∼= B. Moreover,objects of T (P•) are projective in B for any choice of B. Therefore, again applyingProposition 2.6, we have that the admissibility of the complex

0 ← B = T (B) ← T (P•)

in B does not depend on the choice of B. It remains to notice that, in view of the above,the latter complex is admissible in B if and only if A is T -acyclic.

Definition 6.2. An R-S-homomorphism f : A → B from an R-⊗-algebra A to an S-⊗-algebra B is called a two-sided relative homological epimorphism if f is an epimorphismof ⊗-algebras and A is acyclic relative to functor (6.1) from Lemma 6.4.

Proposition 6.5. Any two-sided relative homological epimorphism f : A → B from anR-⊗-algebra A to an S-⊗-algebra B is a left and a right relative homological epimorphism.

Proof. Fix a projective resolution

(6.2) 0 ← A ← P•

in (A, R)-mod-(A, R). In view of Proposition 2.6, the augmented complex (6.2) is splitin R-mod-A. Therefore, for any X ∈ Ob ((B, S)-mod)

(6.3) 0 ← X ← P• ⊗A X

is a projective resolution of X in (A, R)-mod. On the other hand, it follows from theassumption that the complex

0 ← B ← Q• = B ⊗A P• ⊗A B

is split in S-mod-B, and therefore the complex

0 ← X ← Q• ⊗B X

is split in S-mod. Taking account of the isomorphism B ⊗A X ∼= X, we have thatthe latter complex is isomorphic to the complex B ⊗A (6.3). Thus B ⊗A (6.3) is splitin S-mod, and therefore the conditions of Theorem 6.3, (4) hold. This shows that fis a left relative homological epimorphism. A similar argument shows that f is a rightrelative homological epimorphism.



Example 6.1. Fix a ⊗-algebra A and set B = S = A, R = C, and f = 1A. Then it iseasy to see that f is a left (resp., right) relative homological epimorphism if and only ifdg A = 0 (resp., dg Aop = 0). On the other hand, f is a two-sided relative homologicalepimorphism if and only if dbA = 0.

The next proposition shows that one-sided relative homological epimorphisms do in-herit some of the properties of two-sided ones.

Proposition 6.6. Suppose f : A → B is an R-S-homomorphism from an R-⊗-algebra Ato an S-⊗-algebra B and that f is a left or right relative homological epimorphism. ThenA is acyclic relative to the functors

T = B ⊗A ( · ) : (A, R)-mod-(A, R) → LCSspl,

Tr = ( · ) ⊗A B : (A, R)-mod-(A, R) → LCSspl,

T = B ⊗A ( · ) ⊗A B : (A, R)-mod-(A, R) → LCSspl.

Proof. Fix a projective resolution P• of A in (A, R)-mod-(A, R). By Proposition 2.6,augmented complex (2.5) is split in R-mod-A. Therefore the complex

(6.4) 0 ← B ← P• ⊗A B

is split in R-mod-B and, in particular, in LCS. This proves that A is Tr-acyclic. A similarargument shows that the complex

(6.5) 0 ← B ← B ⊗A P•

is split in B-mod-R and, in particular, in LCS, whence the T-acyclicity of A. Further-more, notice that (6.4) (resp., (6.5)) without the term on the left is a projective resolutionof B in (A, R)-mod (resp., in mod-(A, R)). Therefore if f is a left relative homologicalepimorphism, then, applying the functor B ⊗A ( · ) to (6.4), we have a complex

(6.6) 0 ← B ← B ⊗A P• ⊗A B,

which is split in S-mod. If f is a right relative homological epimorphism, then, apply-ing the functor ( · ) ⊗A B to (6.5), we have that (6.6) is split in mod-S. Thus in bothcases (6.6) is split in LCS, which means that A is T -acyclic. Remark 6.2. In fact, the proof of the preceding proposition reduces to the analysis ofthe diagram

(A, A)-mod-(A, R)T

(exact) mod-(A, R)

Tr mod-(B, S)

(exact)

(A, R)-mod-(A, R)

(A-acyclic)

(A-acyclic)

T LCSspl

(A, R)-mod-(A, A)Tr

(exact) (A, R)-mod

T (B, S)-mod

(exact)

Here the vertical arrows are functors acting identically on objects and morphisms. Start-ing with an object A ∈ (A, R)-mod-(A, R) and moving along the arrows of the diagram,at each step we have an object which is acyclic relative to the preceding arrow. Moreover,the right half of the top row corresponds to the case when f is a right relative homologicalepimorphism, and the right half of the bottom row, to the case when f is a left relativehomological epimorphism.

The next theorem shows that in the case R = S = C the possible distinctions betweenleft, right, and two-sided homological epimorphisms disappear.



Theorem 6.7. item Suppose f : A → B is a homomorphism of ⊗-algebras. The followingconditions are equivalent:

(1) Rf• : Db(B-mod) → Db(A-mod) is fully faithful;(1′) Rf• : Db(mod-B) → Db(mod-A) is fully faithful;(2) for any X, Y ∈ Ob (B-mod) and any n ∈ Z+ the canonical morphism

Ext nB(X, Y ) → Ext n

A(X, Y )

is bijective;(2′) for any X, Y ∈ Ob (mod-B) and any n ∈ Z+ the canonical morphism

Ext nB(X, Y ) → Ext n

A(X, Y )

is bijective;(3) for any Y ∈ Ob (B-mod) the canonical morphism Y → Ah(B, Y ) is bijective,

and Ext nA(B, Y ) = 0 for all n ≥ 1;

(3′) for any Y ∈ Ob (mod-B) the canonical morphism Y → hA(B, Y ) is bijective,and Ext n

A(B, Y ) = 0 for all n ≥ 1;(4) for any X ∈ Ob (Db(B-mod)) the canonical morphism

B ⊗L

A(Rf•(X)) → X

is an isomorphism in Db(B-mod);(4′) for any X ∈ Ob (Db(mod-B)) the canonical morphism

(Rf•(X)) ⊗LA B → X

is an isomorphism in Db(mod-B);(5) f is an epimorphism of ⊗-algebras and, furthermore, any X ∈ Ob (B-mod), as

an object of A-mod, is acyclic relative to the functor

B ⊗A ( · ) : A-mod → B-mod;

(5′) f is an epimorphism of ⊗-algebras and, furthermore, any X ∈ Ob (mod-B), asan object of mod-A, is acyclic relative to the functor

( · ) ⊗A B : mod-A → mod-B;

(6) f is an epimorphism of ⊗-algebras and, furthermore, B, as an object of A-mod,is acyclic relative to the functor

B ⊗A ( · ) : A-mod → B-mod;

(6′) f is an epimorphism of ⊗-algebras and, furthermore, B, as an object of mod-A,is acyclic relative to the functor

( · ) ⊗A B : mod-A → mod-B;

(7) f is an epimorphism of ⊗-algebras and, furthermore, A, as an object of A-mod-A,is acyclic relative to the functor

B ⊗A ( · ) ⊗A B : A-mod-A → B-mod-B.

Proof. The implications (1) ⇐⇒ (2) ⇐⇒ (4) ⇐⇒ (5) and (1′) ⇐⇒ (2′) ⇐⇒ (4′) ⇐⇒ (5′)follow from Theorem 6.3.

For (5) ⇐⇒ (5′) ⇐⇒ (6) ⇐⇒ (6′) ⇐⇒ (7), see [48, 3.2].(2) =⇒ (3) and (2′) =⇒ (3′) are obvious.(3) =⇒ (6). As the functors B ⊗A ( · ) and f• are adjoint, it follows from (3), that for

any Y ∈ Ob (B-mod) we have natural isomorphisms

Bh(B, Y ) ∼= Y ∼= Ah(B, Y ) ∼= Bh(B ⊗A B, Y ).



It is easy to see that the composition of these isomorphisms is induced by the canon-ical morphism B ⊗A B → B. Therefore it is an isomorphism in B-mod, i.e., f is anepimorphism of ⊗-algebras. Now we fix a projective resolution

0 ← B ← P•

in A-mod. It follows from (3) that for any Y ∈ Ob (B-mod) the complex

0 → Ah(B, Y ) → Ah(P•, Y )

is exact. On the other hand, it is isomorphic to the complex

0 → Bh(B ⊗A B, Y

)→ Bh

(B ⊗A P•, Y

),

which is exact for any Y ∈ Ob (B-mod) if and only if the complex

0 ← B ⊗A B ← B ⊗A P•

is split in B-mod. Therefore, (6) holds. The implication (3)′ =⇒ (6)′ is proved simi-larly.

Definition 6.3. A ⊗-algebra homomorphism f : A → B is called a homological epimor-phism if it satisfies the equivalent conditions of Theorem 6.7. In this case we shall say(following [46]) that B is stably flat over A.

Remark 6.3. Theorem 6.7 is a “locally convex analog” of a purely algebraic Theorem 4.4from [17]. We decided to give a detailed proof, since the proof in [17] does not carry oververbatim to the case of ⊗-algebras. Moreover, [17] does not mention conditions (4), (4′)and (7), and conditions (5), (5′), (6), and (6′) are stated in terms of the functor Tor ,which is not enough in the ⊗-case.

Remark 6.4. If we state condition (7) of Theorem 6.7 in an explicit form, then we arriveat the following definition: a homomorphism A → B is a homological epimorphism ifand only if for some (or, equivalently, for any) projective resolution

0 ← A ← P0 ← P1 ← · · ·of the bimodule A in A-mod-A the complex

0 ← B ← B ⊗A P0 ⊗A B ← B ⊗A P1 ⊗A B ← · · ·is split in LCS. Thus a homological epimorphism is the same thing as the absolutelocalization in Taylor’s terminology [67].

For examples of homological epimorphisms the reader is referred to [67, 68, 37, 26, 17,46, 16, 15, 48, 44] (see also the introduction to the present paper). A basic example, ob-tained by Taylor [67, 4.3], is the embedding of the polynomial algebra C[x1, . . . , xn] withthe strongest locally convex topology in the Frechet algebra O(Cn) of entire functions.Other examples will be given in Section 9 of this paper (see Theorem 9.12).

7. Relatively quasi-free algebras

Throughout this section R is a fixed ⊗-algebra. Suppose A is an R-⊗-algebra. By anR-extension of A we shall understand an arbitrary open homomorphism of R-⊗-algebrasσ : A → A, where A is an R-⊗-algebra. As usual, an extension will written as a sequence

(7.1) 0 → Ii−→ A

σ−→ A → 0,

where I = Kerσ and i is the tautological embedding. Notice that (7.1) is a topo-logically exact (see the Appendix, Example 10.5) sequence of R-⊗-bimodules. Exten-sion (7.1) is said to be R-admissible (henceforth, simply admissible) if this sequence issplit in R-mod-R, i.e., if σ has a right inverse morphism of R-⊗-bimodules. If σ has a right



inverse morphism of R-⊗-algebras, then extension (7.1) is said to be R-split (henceforth,simply split).

Recall that extension (7.1) is said to be singular if I2 = 0. In this case I becomes anA-⊗-bimodule when, for a ∈ A and x ∈ I, we set a · x = ux and x · a = xu for arbitraryu ∈ σ−1(a). If X is an A-⊗-bimodule, then an R-extension of A by X is, by definition,a topologically exact sequence

(7.2) 0 → Xκ−→ A

σ−→ A → 0,

in which A is an R-⊗-algebra, σ is a homomorphism of R-⊗-algebras, and κ is a morphismof A-⊗-bimodules. Clearly, each R-extension of A by X gives rise to a singular R-extension of the form (7.1), and conversely, as follows from the above, any singularextension can be obtained this way; moreover, κ establishes an isomorphism between Xand I in A-mod-A.

Two R-extensions 0 → X → A → A → 0 and 0 → X → A′ → A → 0 of A by X aresaid to be equivalent if there is an isomorphism of R-⊗-algebras ϕ : A → A′ such thatthe diagram

A

ϕ

X

A

A′

is commutative. As in the case R = C, it is not difficult to check that the set of equiva-lence classes of admissible R-extensions of A by X is in one-to-one correspondence withH2(A, R; X); moreover the zero element of H2(A, R; X) corresponds to the equivalenceclass of a split extension.

Proposition 7.1. The following properties of the R-⊗-algebra A are equivalent:(i) dbR A ≤ 1;(ii) H2(A, R; X) = 0 for any X ∈ A-mod-A;(iii) any admissible singular R-extension of A is split;(iv) for any R-⊗-algebra B, any admissible singular R-extension 0 → I → A → B →

0, and any R-homomorphism ϕ : A → B there is an R-homomorphism ψ : A → A

making the diagram

A

ϕ

ψ

0 I A B 0

commute.

Proof. Clearly, (i)⇐⇒(ii) and (iv)=⇒(iii); furthermore, it follows from the above that(ii)⇐⇒(iii). To prove the implication (iii)=⇒(iv), take an arbitrary admissible singularR-extension 0 → I

i−→ Aσ−→ B → 0 and consider its preimage under ϕ:

0 Ii′ A ×B A

σ′

π

A

ϕ

0

0 Ii A

σ B 0

Here A ×B A = (u, a) ∈ A × A : σ(u) = ϕ(a) (it is easy to see that this is an R-⊗-algebra), i′(a) = (i(a), 0), σ′(u, a) = a and π(u, a) = u. The top row of this diagram is anadmissible singular R-extension of A (its admissibility follows from Axiom Ex2 applied



to the exact category (R-mod-R)spl, see the Appendix, but it is also easy to check itdirectly). By (iii), there is an R-homomorphism ψ′ : A → A×B A which is a right inverseof σ′. It remains to set ψ = πψ′.

Definition 7.1. An R-⊗-algebra A, satisfying the equivalent conditions of Proposi-tion 7.1 is said to be relatively quasi-free. If R = C, then A is said to be quasi-free.

To check whether an algebra is quasi-free it is convenient to use the bimodule of non-commutative differential 1-forms [7]. It is defined as follows. Fix an R-⊗-algebra A andconsider the functor Der R(A, · ) from A-mod-A to the category of sets. As was shownby Cuntz and Quillen [7] (in a purely algebraic situation), this functor is representable.Its representing object is denoted Ω1

RA and is called the bimodule of relative differen-tial 1-forms of A. In other words, the bimodule of relative differential 1-forms is a pair(Ω1

RA, dA) consisting of an A-⊗-bimodule Ω1RA and an R-derivation dA : A → Ω1

RA suchthat for any A-⊗-bimodule X and any R-derivation D : A → X there exists a uniquemorphism of A-⊗-bimodules ϕ : Ω1

RA → X making the diagram

(7.3)

Ω1RA

ϕ

A

dA

D

X

commute. The derivation dA is called the universal R-derivation of A.To construct Ω1

RA, denote the completion of the quotient space A/Im ηA by A; clearly,this is an R-⊗-bimodule. The image of a ∈ A under the canonical map A → A will bedenoted a. As a left A-⊗-module, Ω1

RA coincides with A ⊗R A. To define a structure ofa right A-⊗-module on it, denote the elementary tensor a0 ⊗ a1 ∈ A ⊗R A by a0da1; thetensor 1da will be simply written as da. It is not difficult to see that there is a uniquecontinuous bilinear map Ω1

RA × A → Ω1RA, (ω, a) → ω · a, such that

(7.4) a0da1 · a = a0d(a1a) − a0a1da.

This map defines a right action of A on Ω1RA and makes it an A-⊗-bimodule. It follows

from (7.4) that the map dA : A → Ω1RA, a → da, is an R-derivation. If D : A → X is an

arbitrary R-derivation of A with values in a bimodule X, then the unique morphism ϕmaking diagram (7.3) commute is given by ϕ(a0da1) = a0 · D(a1); it is straightforwardto check that ϕ is a morphism of bimodules. Thus, the pair (Ω1

RA, dA) is indeed thebimodule of relative differential 1-forms.

Consider the sequence of A-⊗-bimodules

(7.5) 0 ← Am←− A ⊗R A

j←− Ω1RA ← 0.

Here m is the multiplication on A, and j is the morphism of bimodules corresponding tothe inner derivation −ad1⊗1 : A → A ⊗R A, a → [1 ⊗ 1, a]. In other words, j is uniquelydetermined by the formula j(a0da1) = a0(1⊗ a1 − a1 ⊗ 1). Clearly, mj = 0, i.e., (7.5) isa complex of A-bimodules.

Proposition 7.2. Sequence (7.5) is split in A-mod-R and R-mod-A.

Proof. Consider the morphisms of A-R-⊗-bimodules,

s : A → A ⊗R A, a → a ⊗ 1;

t : A ⊗R A → Ω1RA, a0 ⊗ a1 → a0da1.

A direct calculation shows that s and t provide a contracting homotopy for (7.5) making itsplit in A-mod-R. Furthermore, define a morphism of R-A-⊗-bimodules s′ : A → A ⊗R A



by setting s′(a) = 1 ⊗ a; then, since ms′ = 1A and j = kerm, we have that (7.5) is alsosplit in R-mod-A.

Proposition 7.3. The R-⊗-algebra A is relatively quasi-free if and only if Ω1RA is projec-

tive in some (equivalently, in any) of the exact categories (A, R)-mod-(A, R),(A, A)-mod-(A, R), and (A, R)-mod-(A, A).

Proof. By Proposition 7.2, the sequence (7.5) is admissible in the exact categories(A, R)-mod-(A, R), (A, A)-mod-(A, R), and (A, R)-mod-(A, A). Moreover, its middleterm is projective in each of these categories. Hence, taking into account Corollary 2.8,for any A-⊗-bimodule X we have isomorphisms

H2(A, R; X) ∼= Ext 1Ae, Re(Ω1

RA, X) ∼= Ext 1Ae, A ⊗Rop(Ω1

RA, X) ∼= Ext 1Ae, R ⊗Aop(Ω1

RA, X).

The rest is clear.

Remark 7.1. Notice that the notion of a relatively quasi-free algebra has an obviouspurely algebraic analog (see [7] for R = C). Everything that was said above aboutrelatively quasi-free ⊗-algebras and noncommutative differential forms carries over easilyto algebras without topology. In that regard we remark that if R is an algebra and A is anR-algebra having a countable dimension as a vector space, then its bimodule of relativedifferential 1-forms, constructed in a purely algebraic situation and then endowed withthe strongest locally convex topology, is isomorphic to the bimodule of relative differentialforms of the strongest locally convex algebra As over Rs (see Proposition 2.3). Combiningthis with Remark 2.2, we have that As is relatively quasi-free over Rs as a ⊗-algebra ifand only if A is relatively quasi-free over R in the purely algebraic sense.

Remark 7.2. Quasi-free algebras admit some other equivalent definitions. In particular,they can be characterized in terms of connections. For details (when R = C), see [7, 9]or [43].

Next, we want to determine the action of the Arens–Michael functor on the bimod-ule Ω1

RA (see Remark 3.8). Consider first a more general situation. Suppose A is anR-⊗-algebra, B is an S-⊗-algebra, and (f, g) is an R-S-homomorphism from A to B.Fix an arbitrary B-⊗-bimodule X and consider the commutative diagram

Der S(B, X) Der R(A, X)

BhB(Ω1SB, X)

AhA(Ω1RA, X) BhB(B ⊗A Ω1

RA ⊗A B, X)

Here the equalities denote canonical isomorphisms, the top arrow was defined in (3.5),and the bottom arrow is uniquely determined by the top one and the commutativity ofthe diagram. Substituting Ω1

SB for X in the bottom row, we see that the morphism ofB-⊗-bimodules B ⊗A Ω1

RA ⊗A B → Ω1SB (this is a noncommutative analog of the pull-

back operation of differential forms), denoted f∗, corresponds to the identity morphismof Ω1

SB.

Proposition 7.4. Let A, R, S be ⊗-algebras and g : R → S an epimorphism of ⊗-algebras. Assume that A is an R-⊗-algebra and that its Arens–Michael envelope A isan S-⊗-algebra such that the pair (ιA, g) is an R-S-homomorphism from A to A. Thenthe morphism

(ιA)∗ : A ⊗A Ω1RA ⊗A A → Ω1

SA

is an isomorphism of A-⊗-bimodules.



Proof. Use Proposition 3.8.

Corollary 7.5. For any R-⊗-algebra A the morphism

(ιA)∗ : A ⊗A Ω1RA ⊗A A → Ω1

RA

is an isomorphism of A-⊗-bimodules.

Theorem 7.6. Let A, R, S be ⊗-algebras and g : R → S an epimorphism of ⊗-algebras.Suppose that A is an R-⊗-algebra and its Arens–Michael envelope A is an S-⊗-algebrasuch that the pair (ιA, g) is an R-S-homomorphism from A to A. Assume also that A isrelatively quasi-free over R. Then:

(i) A is relatively quasi-free over S;(ii) ιA : A → A is a two-sided relative homological epimorphism.

Proof. Assertion (i) follows from Propositions 7.3 and 7.4. The same propositions implythat (7.5) is a projective resolution of A in (A, R)-mod-(A, R). Applying to it the functorA ⊗A ( · ) ⊗A A and using Propositions 6.1 (iv) and 7.4, we have a sequence

0 ← Am←− A ⊗S A

j←− Ω1SA ← 0,

which is admissible in (A, S)-mod-(A, S) by Proposition 7.2. The rest is clear.

Corollary 7.7. Suppose A is a relatively quasi-free R-⊗-algebra. Then A is relativelyquasi-free over R and ιA : A → A is a two-sided relative homological epimorphism.

Remark 7.3. Setting in the preceding corollary R = C and A = Fn we recover Taylor’sresult [67] that the embedding of the free algebra Fn in its Arens–Michael envelope is ahomological epimorphism.

Now we want to give several examples of relatively quasi-free algebras.

Proposition 7.8. Suppose R is an Arens–Michael algebra and A is any of the followingtwo R-⊗-algebras:

(i) A = O(C, R; α, δ), where α : R → R is an endomorphism and δ : R → R is anα-derivation such that the family α, δ is m-localizable;

(ii) A = O(C×, R; α), where α : R → R is an automorphism such that the familyα, α−1 is m-localizable.

Then the bimodule Ω1RA is isomorphic to Aα ⊗R A, and the universal derivation

d : A → Aα ⊗R A

is uniquely determined by the formula d(z) = 1 ⊗ 1. Consequently, A is relatively quasi-free.

Proof. Consider first case (i). In view of (2.3) and Remark 4.6, for any A-⊗-bimodule Xwe have the canonical bijections

Der R(A, X) ∼=ϕ ∈ Hom R-AM(A, A × X) : p1ϕ = 1A

∼=b ∈ A × X : p1(b) = z, b · r = α(r) · b + δA×X(r) ∀r ∈ R

∼=x ∈ X : (zr, x · r) = (α(r)z, α(r) · x) + (δ(r), 0) ∀r ∈ R

=x ∈ X : x · r = α(r) · x ∀r ∈ R

∼= RhR(Rα, X)∼= AhA(A ⊗R Rα ⊗R A, X) ∼= AhA(Aα ⊗R A, X).

It is easy to see that these bijections give rise to an isomorphism of functors of X, whencethe first assertion. To prove the second assertion, we set X = Aα ⊗R A and notice that



the R-derivation d corresponding to the identity morphism 1X under the above bijectionssends z ∈ A to 1 ⊗ 1 ∈ X.

To prove (ii), notice that for any A-⊗-bimodule X and any x ∈ X the element (z, x) ∈A×X is invertible and its inverse is (z−1,−z−1 · x · z−1). In view of this and (4.26), wecan finish the argument as in case (i). Proposition 7.9. Suppose R is an algebra of countable dimension and A is any of thefollowing two R-algebras:

(i) A = R[t; α, δ], where α : R → R is an endomorphism and δ : R → R is anα-derivation;

(ii) A = R[t, t−1; α], where α : R → R is an automorphism.Endow R and A with the strongest locally convex topology. Then the bimodule Ω1

RAis isomorphic to Aα ⊗R A (which, in turn, is isomorphic to Aα ⊗R A endowed withthe strongest locally convex topology), and the universal derivation d : A → Aα ⊗R A isuniquely determined by the formula d(t) = 1⊗1. Consequently, A is relatively quasi-free.

The proof of this proposition is similar to that of Proposition 7.8 and is thereforeomitted. We also remark that the assumption that R is of countable dimension wasneeded only to guarantee that the multiplications on R and A are jointly continuous.A purely algebraic analog of the preceding proposition is, of course, true without anyrestrictions on the dimension of R. The same applies to Proposition 7.11 below.

The following two propositions are proved exactly the same way as Proposition 2.9of [7]. We omit the proofs.

Proposition 7.10. Suppose R is an Arens–Michael algebra, M is an R-⊗-bimodule, andA = TR(M). Then the bimodule Ω1

RA is isomorphic to A ⊗R M ⊗R A, and the universalderivation d : A → A ⊗R M ⊗R A is uniquely determined by the formula d(m) = 1⊗m⊗1for m ∈ M. As a consequence, A is relatively quasi-free.

Proposition 7.11. Suppose R is a countably dimensional algebra, M is a countablydimensional R-bimodule, and A = TR(M). Endow R, M , and A with the strongest locallyconvex topology. Then the bimodule Ω1

RA is isomorphic to A ⊗R M ⊗R A (which, in turn,is isomorphic to A⊗R M ⊗R A endowed with the strongest locally convex topology), andthe universal derivation d : A → A ⊗R M ⊗R A is uniquely determined by the formulad(m) = 1 ⊗ m ⊗ 1 for m ∈ M. Consequently, A is relatively quasi-free.

8. Finiteness conditions

Definition 8.1. Suppose A is a topological algebra and X is a left topological A-module.We shall say that X is strictly finitely generated if for some n ∈ N there is an openmorphism from An onto X.

Remark 8.1. It is not difficult to see that if A is locally convex and X is finitely gener-ated and endowed with the strongest locally convex topology, then it is strictly finitelygenerated.

Remark 8.2. If A is a Frechet algebra, then it follows easily from the Banach OpenMapping Theorem that a left topological A-module X is strictly finitely generated if andonly if it is finitely generated and is Frechet.

Remark 8.3. If A → B is a homomorphism of Frechet algebras and X is a finitelygenerated left Frechet A-module, then the left Frechet B-module B ⊗A X is also finitelygenerated. This follows from the preceding remark and the fact that the functor oftaking projective tensor product with a Frechet module sends open morphisms of Frechetmodules to open morphisms (see [25, II.4.12]).



Proposition 8.1. Suppose A is a topological algebra and X is a strictly finitely generatedleft topological A-module. Suppose that A is metrizable. Then the A-module X is strictlyfinitely generated.

Proof. We may assume that X = An/Y for some submodule Y ⊂ An. It follows fromProposition 3.12 and Remark 3.7 that X = An/ιnA(Y ). The rest is clear.

Suppose A is an R-⊗-algebra and P is an A-⊗-bimodule. Recall that P is projectivein (A, R)-mod-(A, R) if and only if P is a retract of a bimodule A ⊗R M ⊗R A, whereM ∈ R-mod-R.

Definition 8.2. We shall say that a projective (A, R)-⊗-bimodule P satisfies the finite-ness condition (f1) if it is a retract of a bimodule A ⊗R M ⊗R A, where M ∈ R-mod-Ris strictly finitely generated as a left R-module.

Definition 8.3. We shall say that a projective (A, R)-⊗-bimodule P satisfies the finite-ness condition (f2) if it is a retract of a bimodule A ⊗R M ⊗R A, where M ∈ R-mod-Ris isomorphic to the R-⊗-bimodule Rn for some n.

Remark 8.4. Clearly, condition (f2) implies condition (f1). If R = C, then both condi-tions are equivalent to P being a retract of (A ⊗A)n for some n. This means preciselythat P is strictly projective (see [27]) and finitely generated in A-mod-A.

Definition 8.4. We shall say that an R-⊗-algebra A is an algebra of (f1)-finite type(resp., (f2)-finite type) if in (A, R)-mod-(A, R) it has a finite projective resolution all ofwhose bimodules satisfy the finiteness condition (f1) (resp., (f2)).

When R = C, the preceding definition means that A has a finite projective resolutionall of whose bimodules are strictly projective and finitely generated. These conditionsare satisfied, for example, by the algebra of polynomials C[t1, . . . , tn] with the strongestlocally convex topology, and also by the algebra O(U) of holomorphic functions on a poly-domain U ⊂ Cn: in both cases the resolution can be chosen to be a Koszul complex [67].Other examples will appear in the next section.

Definition 8.5. We shall say that an R-⊗-algebra A is relatively (f1)-quasi-free (resp.,relatively (f2)-quasi-free) if Ω1

RA satisfies condition (f1) (resp., (f2)).

It follows immediately from Propositions 7.2 and 7.3 that a relatively (f1)-quasi-free(resp., relatively (f2)-quasi-free) algebra is quasi-free in the sense of Definition 7.1 andis an algebra of (f1)-finite type (resp., (f2)-finite type).

Remark 8.5. Notice that Definitions 8.3, 8.4, and 8.5 have obvious purely algebraicanalogs (cf. Remark 7.1). For example, the R-algebra A is relatively (f1)-quasi-free if thebimodule Ω1

RA constructed in a purely algebraic situation is a retract of the A-bimoduleA⊗R M ⊗R A, where the R-bimodule M is finitely generated as a left R-module. Itis not difficult to see that if we restrict ourselves to countably dimensional algebrasand modules, then the “algebraic” finiteness conditions (i.e., conditions (f1) and (f2) forbimodules, the property of being an algebra of (f1)- or (f2)-finite type, or (f1)-quasi-free,or (f2)-quasi-free) are equivalent to the corresponding “topological” finiteness conditionsfor the same algebras and bimodules endowed with the strongest locally convex topology.We skip the verification of all these assertions since this is not difficult and reduces toapplying Proposition 2.3.

Example 8.1. The algebras O(C, R; α, δ) and O(C×, R; α) of Proposition 7.8 are rel-atively (f1)-quasi-free. The same is true for the algebras R[t; α, δ] and R[t, t−1; α] ofProposition 7.9. If α = 1R, then the above algebras are relatively (f2)-quasi-free.



Example 8.2. The algebras TR(M) and TR(M) of Propositions 7.10 and 7.11 are rel-atively (f1)-quasi-free if M is strictly finitely generated in R-mod, and relatively (f2)-quasi-free if M is a retract of Rn in R-mod-R for some n ∈ N .

Lemma 8.2. Suppose B is an Arens–Michael algebra and N ∈ B-mod is a strictlyfinitely generated module. Then the algebra Bh(N) endowed with the topology of simpleconvergence is Arens–Michael.

Proof. If N = Bn, then Bh(N) is topologically isomorphic to Mn(B) and is thereforeArens–Michael (see Section 2). In the general case, for some n ∈ N there is a openmorphism σ : Bn → N . Set K = Ker σ and C = ϕ ∈ Bh(Bn) : ϕ(K) ⊂ K. Clearly, Cis a closed subalgebra of Bh(Bn) and is therefore Arens–Michael. It is also clear that toeach ϕ ∈ C there corresponds a unique ψ ∈ Bh(N), acting by the rule ψ(σ(x)) = σ(ϕ(x))(x ∈ Bn). Moreover, the map θ : C → Bh(N), ϕ → ψ, is a homomorphism of algebras.

The morphism σ induces two continuous linear maps

σ∗ : Bh(N, N) → Bh(Bn, N), ψ → ψ σ,

σ∗ : Bh(Bn, Bn) → Bh(Bn, N), ϕ → σ ϕ.

Obviously, σ∗ is injective and

Im σ∗ =ϕ ∈ Bh(Bn, N) : ϕ(K) = 0

.

Therefore Im σ∗ is closed. Moreover, a simple verification shows that for any finite setD ⊂ Bn and any neighborhood of zero U ⊂ N we have

σ∗(M(σ(D), U))

= M(D, U) ∩ Im σ∗

(see (2.1) for the notation). Since any finite subset in N is an image of some finite subsetof Bn, we have that σ∗ is topologically injective, i.e., it is a topological isomorphismbetween Bh(N, N) and Imσ∗. The inverse isomorphism will be denoted κ. Notice thatthe space Bh(N, N) is complete, because it is isomorphic to a closed subspace Imσ∗ ofthe complete space Bh(Bn, N) ∼= Nn.

Furthermore, it is not difficult to see that σ∗ is open. Indeed, under the naturalisomorphism Bh(Bn, X) ∼= Xn (X ∈ B-mod) it is sent to σn, which is open as so is σ.

It follows from the definition of the algebra C that C = σ−1∗ (Im σ∗). Since the restric-

tion of an open map to the preimage of any set is open, we have that σ∗|C : C → Im σ∗

is open. It follows from the definition of the homomorphism θ that θ = κ σ∗|C ; hence θis continuous and open. Therefore, Bh(N) is isomorphic to the quotient algebra C/Ker θand is therefore locally m-convex. In view of the aforementioned completeness of Bh(N),this concludes the proof.

Lemma 8.3. Suppose R is a ⊗-algebra whose Arens–Michael envelope S is metrizable.Let B be an Arens–Michael algebra and N a Frechet B-R-⊗-bimodule which is strictlyfinitely generated in B-mod. Then N admits a structure of a right S-module, whichextends the original structure of a right R-module (i.e., such that n · ιR(r) = n · r for alln ∈ N and r ∈ R) and makes it a Frechet B-S-⊗-bimodule.

Proof. Consider the algebra Bh(N) and endow it, as in the preceding lemma, with thetopology of simple convergence. Clearly, the map

m : R → Bh(N)op, m(r)(n) = n · ris an algebra homomorphism and it is not difficult to check that it is continuous. Indeed,for any neighborhood of zero V ⊂ N and any finite subset D ⊂ N there is a neighborhoodof zero U ⊂ R such that D ·U ⊂ V, which is equivalent to the inclusion m(U) ⊂ M(D, V ).Therefore m is continuous. By the preceding lemma, Bh(N) is Arens–Michael; hence



there exists a unique continuous homomorphism m : S → Bh(N)op such that mιR = m.Setting n · s = m(s)(n) for s ∈ S, n ∈ N, we have a structure of a right S-module on Nextending the original structure of a right R-module and making N a B-S-bimodule.Clearly, the constructed action N × S → N is separately continuous. But S and Nare Frechet spaces; hence this action is jointly continuous. Therefore N is a FrechetB-S-⊗-bimodule.

Corollary 8.4. Suppose R is a ⊗-algebra whose Arens–Michael envelope S is metrizable,and a bimodule M ∈ R-mod-R is strictly finitely generated as a left R-module. Then theS-⊗-bimodule M = S ⊗R M ⊗R S is isomorphic to S ⊗R M in S-mod and, in particular,is strictly finitely generated in S-mod.

Proof. Consider the S-R-⊗-bimodule N = S ⊗R M. It follows from Propositions 8.1and 3.11 that it is strictly finitely generated in S-mod and, in particular, is a Frechetmodule. On the other hand, it follows from Lemma 8.3 that on N there is a structureof a right S-⊗-module extending the structure of a right R-⊗-module and making it anS-⊗-bimodule. As the canonical image of R in S is dense, we have the isomorphisms ofS-⊗-bimodules,

M = N ⊗R S ∼= N ⊗S S ∼= N.

But, as we mentioned before, N is strictly finitely generated in S-mod.

Corollary 8.5. Suppose A, R, S are ⊗-algebras and g : R → S is an epimorphism of⊗-algebras. Suppose that on A we have a structure of an R-⊗-algebra, and on its Arens–Michael A envelope we have a structure of an S-⊗-algebra such that the pair (ιA, g) isan R-S-homomorphism from A to A. Assume also that A is relatively quasi-free over R.Then:

(i) If A is relatively (f2)-quasi-free over R, then A is relatively (f2)-quasi-free over S;(ii) If (S, g) = (R, ιR), S is metrizable and A is relatively (f1)-quasi-free over R,

then A is relatively (f1)-quasi-free over S.

Proof. Suppose Ω1RA is a retract of A ⊗R M ⊗R A in A-mod-A. By Proposition 7.4,

the A-⊗-bimodule Ω1SA is isomorphic to A ⊗A Ω1

RA ⊗A A and is therefore a retract ofA ⊗R M ⊗R A, which, in turn, is isomorphic to A ⊗S M ⊗S A.

If M ∼= Rn for some n ∈ N, then M ∼= (S ⊗R S)n ∼= Sn. In view of the above, thisproves (i). To prove (ii), apply Corollary 8.4.

Corollary 8.6. If A is a relatively (f2)-quasi-free R-⊗-algebra, then A is relatively (f2)-quasi-free over R. If A is a relatively (f1)-quasi-free R-⊗-algebra and R is metrizable,then A is relatively (f1)-quasi-free over R.

9. Theorems on stable flatness of Arens–Michael envelopes

In this section we shall show, for some finitely generated algebras A, that the canonicalhomomorphism ιA from A to its Arens–Michael envelope A is a homological epimorphism,i.e., A is stably flat over A (see Definition 6.3). Our main technical tool will be the fol-lowing theorem, establishing a connection between the notions of relative and “absolute”homological epimorphism.

Theorem 9.1. Suppose (f, g) : (A, R) → (B, S) is an R-S-homomorphism from an R-⊗-algebra A to an S-⊗-algebra B. Assume that g is a homological epimorphism and fis a left or right relative homological epimorphism. Also assume that A is projective



in R-mod and B is projective in S-mod, and that one of the following two conditions issatisfied:

(i) S and B are Frechet–Arens–Michael algebras such that S = R, g = ιR, and A isan algebra of (f1)-finite type;

(ii) A is an algebra of (f2)-finite type.

Then f is a homological epimorphism.

For the proof we need the following lemma.

Lemma 9.2. Let A, B be exact categories and F : A → B an additive covariant functorhaving a left derived functor. Suppose X → Y → Z is an admissible pair in A andthat F sends it to an admissible pair in B. Suppose also that two objects from X, Y, Zare F -acyclic. Then so is the third.

Proof. The admissible pairs X → Y → Z and FX → FY → FZ give rise to distin-guished triangles X → Y → Z → X[1] and FX → FY → FZ → (FX)[1] in D−(A) andD−(B), respectively (see [32, 11.6]). Applying the functor LF to the first one, we have adistinguished triangle in D−(B) which maps to the second one:

LF (X)

LF (Y )

LF (Z)

LF (X)[1]

FX FY FZ (FX)[1]

By the assumptions, the first two vertical arrows are isomorphisms in D−(B). Hence sois the third one (see Proposition 10.1).

Corollary 9.3. Suppose A, B and F are the same as in the preceding lemma. Assumefurthermore that F sends admissible epimorphisms to epimorphisms. Let

0 ← X0 ← X1 ← · · · ← Xn ← 0

be an admissible complex in A which is sent by F to an admissible complex in B andwhose objects X1, . . . , Xn are F -acyclic. Then X0 is also F -acyclic.

Proof. When n = 1, there is nothing to prove, and when n = 2 everything follows fromLemma 9.2. Suppose that the assertion is true for complexes of length at most n, wheren ≥ 2. We shall show that it is true for complexes of length n + 1. By the assumption,the complexes

0 ← X0d0←− · · · dn−2←−−− Xn−1

dn−1←−−− Xndn←− Xn+1 ← 0,

0 ← FX0Fd0←−− · · · Fdn−2←−−−− FXn−1

Fdn−1←−−−− FXnFdn←−−− FXn+1 ← 0

are admissible. Setting K = Ker dn−2 and L = Ker Fdn−2, we have admissible complexes

0 ← X0d0←− X1

d1←− · · · dn−2←−−− Xn−1 ← K ← 0,(9.1)

0 ← K ← Xndn←− Xn+1 ← 0,(9.2)

0 ← FX0Fd0←−− FX1

Fd1←−− · · · Fdn−2←−−−− FXn−1 ← L ← 0,

0 ← L ← FXnFdn←−−− FXn+1 ← 0.

Since they are admissible, we have, in particular, that K = Coker dn, and L = CokerFdn.



Consider the diagram

· · · FXn−1Fdn−2 FXn

Fdn−1

δ

β

· · ·Fdn

L

α

v FK

γ

u

Here α = kerFdn−2, β = F (coker dn), γ = F (ker dn−2), δ = coker Fdn, and u and v areuniquely determined by the conditions αu = γ and vδ = β.

We haveαuvδ = γβ = Fdn−1 = αδ.

Since α is a monomorphism and δ is an epimorphism, we also have uv = 1L. Furthermore,the assumption on F and the admissibility of (9.2) imply that β is an epimorphism. Asvδ = β, we have that v is also an epimorphism. This and the equality uv = 1L showthat u and v are inverses of each other and thus FK ∼= L. To finish the proof, apply theinduction assumption to (9.1) and (9.2).

Proof of Theorem 9.1. Consider the commutative diagram

(9.3)

mod-A( · ) ⊗A B mod-B

mod-R

( · ) ⊗R A

( · ) ⊗R S

mod-S

( · ) ⊗S B

We need to show that B ∈ mod-A is acyclic relative to ( · ) ⊗A B.Choose a projective resolution

(9.4) 0 ← A ← P0 ← P1 ← · · · ← Pn ← 0

of A in (A, R)-mod-(A, R) such that all the bimodules Pi satisfy condition (f1) in case (i)(resp., condition (f2) in case (ii)). By Proposition 6.6, the complexes

0 ← B ← B ⊗A P0 ← B ⊗A P1 ← · · · ← B ⊗A Pn ← 0,(9.5)

0 ← B ← B ⊗A P0 ⊗A B ← B ⊗A P1 ⊗A B ← · · · ← B ⊗A Pn ⊗A B ← 0(9.6)

are split in LCS. Notice that (9.6) can be obtained from (9.5) by applying the functor( · ) ⊗A B : mod-A → mod-B. Since the epimorphisms in these categories are exactly themorphisms with dense images, it is easy to see that this functor sends epimorphisms toepimorphisms. Therefore, in view of Corollary 9.3, to finish the proof it suffices to showthat the modules B ⊗A Pi (i = 0, . . . , n) are acyclic relative to this functor.

Fix an arbitrary i ∈ 0, . . . , n. Without loss of generality we may assume thatPi = A ⊗R M ⊗R A, where M ∈ R-mod-R and, in addition, that M is strictly finitelygenerated in R-mod in case (i), and that M is isomorphic to Rm in R-mod-R in case (ii).

Now we shall show that the structure of a right R-⊗-module on the B-R-⊗-bimoduleN = B ⊗R M extends to a structure of a right S-⊗-module. Indeed, in case (ii), N ∼= Bm

in B-mod-R and the assertion is true for obvious reasons. In case (i), M is strictly finitelygenerated in R-mod; hence S ⊗R M is strictly finitely generated in S-mod (see Propo-sition 8.1) and therefore N = B ⊗R M = B ⊗S (S ⊗R M) is strictly finitely generatedin B-mod (see Remark 8.3). Thus, applying Lemma 8.3, we have that the structure of aright R-⊗-module on N extends to a structure of a right S-⊗-module.

Combining this result with the fact that g is a homological epimorphism, we see that Nis acyclic relative to the bottom arrow in (9.3). On the other hand, since A ∈ R-mod



and B ∈ S-mod are projective, the vertical arrows in that diagram are exact functors.Therefore, by Corollary 2.5, the right A-⊗-module N ⊗R A = B ⊗A Pi is acyclic relativeto the top arrow in (9.3). In view of the above, this implies that f is a homologicalepimorphism.

Henceforth all finitely generated algebras and finitely generated modules over themwill be assumed to be endowed with the strongest locally convex topology.

Corollary 9.4. Let R be a finitely generated algebra and A a finitely generated relatively(f1)-quasi-free R-algebra which is projective as a left R-module. Suppose that A is pro-jective in R-mod. If ιR : R → R is a homological epimorphism, then ιA : A → A is alsoa homological epimorphism.

Proof. Combine Corollary 7.7 with Theorem 9.1.

Corollary 9.5. Let R be a finitely generated algebra and M an R-bimodule which isfinitely generated and projective as a left R-module. Suppose that TR(M) is projectivein R-mod. If ιR : R → R is a homological epimorphism, then ιTR(M) : TR(M) → TR(M)is also a homological epimorphism.

Proof. Notice that since M is projective as a left R-module, TR(M) is projective as aleft R-module. Now use Corollary 9.4 and Example 8.2.

Remark 9.1. When R = C, the preceding corollary is Taylor’s result [67] that the embed-ding of the free algebra Fn in its Arens–Michael envelope is a homological epimorphism(see Remark 4.9 and Remark 7.3).

Remark 9.2. In regard to the proof of the preceding corollary, we remark that the R-bimodule M = R ⊗R M ⊗R R is isomorphic to R ⊗R M in R-mod (see Corollary 8.4)and, therefore, is projective in R-mod. By induction, this easily implies that all its tensorpowers M ⊗n (see Section 4.2) are also projective in R-mod. Since the analytic tensoralgebra TR(M) is the completion of the direct sum of the bimodules M ⊗n, it is natural toask whether it is projective in R-mod, i.e., whether one of the conditions of the precedingcorollary is redundant. Unfortunately, the analytic tensor algebra has the followingdrawback, which is not present for the algebraic tensor algebra: if S is an Arens–Michaelalgebra and N is an S-⊗-bimodule, then the projectivity of N in S-mod (or in mod-S)does not, in general, imply that the analytic tensor algebra TS(N) is projective in S-mod(resp., in mod-S). In particular, if α is an m-localizable automorphism S, then theanalytic Ore extension O(C, S; α) = St; α = TS(Sα) (see Propositions 4.9 and 4.11),being a projective (and even free) left S-⊗-module need not be a projective right S-⊗-module. In fact, one can show that the algebra of holomorphic functions on the quantumplane Oq(C2) (see Corollary 5.14) with |q| > 1 is not projective as a right module overits subalgebra generated by the element x (in contrast with a purely algebraic situation).We will not prove this assertion, since we do not need it for our purposes.

Corollary 9.6. Let R be a finitely generated algebra, α : R → R an endomorphism, andδ : R → R an α-derivation. Suppose that their canonical extensions α, δ to R form an m-localizable family. If ιR : R → R is a homological epimorphism, then ιR[t;α,δ] : R[t; α, δ] →O(C, R; α, δ) is also a homological epimorphism.

Proof. Set A = R[t; α, δ]. By Theorem 5.17, we have

A ∼= O(C, R; α, δ).



This implies, in particular, that A is projective (and even free) in R-mod. In view ofExample 8.1, to finish the proof it suffices to apply Corollary 9.4. Corollary 9.7. Let R be a finitely generated algebra, α : R → R an automorphism, andα its canonical extension to R. Suppose that the family α, α−1 is m-localizable. IfιR : R → R is a homological epimorphism, then ιR[t,t−1;α] : R[t, t−1; α] → O(C×, R; α) isalso a homological epimorphism.

Proof. In view of Theorem 5.21, the proof is similar to that of the preceding corollary. Notice that in Corollaries 9.4–9.7 we dealt with the situation described in Theorem

9.1, (i). We shall now examine part (ii).

Corollary 9.8. Let R be a finitely generated algebra and δ : R → R a derivation. Ifj : R → Rδ is a homological epimorphism, then ιR[t;δ] : R[t; δ] → O(C, Rδ; δ) is also ahomological epimorphism.

Proof. The proof is based on Theorem 5.1 and Theorem 9.1, (ii) and is similar to theproof of Corollary 9.6.

To apply these results to the concrete algebras from Section 5, we will need some factsabout homological epimorphisms of polynomial algebras.

An augmented ⊗-algebra will be understood as a ⊗-algebra A with a homomorphismεA : A → C. Homomorphisms between augmented ⊗-algebras are defined in an obviousway. If A is an augmented ⊗-algebra, then we consider C as an A-⊗-module obtainedby a change of scalars along εA.

Definition 9.1. Suppose f : A → B is a homomorphism of augmented ⊗-algebras. Weshall say that f is a weak homological epimorphism if the canonical morphism B ⊗L

A C →C is an isomorphism in Db(B-mod).

Remark 9.3. Weak homological epimorphisms were introduced in [48] under the nameof “weak localization”. Notice that f : A → B is a weak homological epimorphism ifand only if C ∈ A-mod is acyclic relative to the functor B ⊗A ( · ) : A-mod → B-mod andthe canonical morphism B ⊗A C → C is an isomorphism in B-mod. This is proved byconsidering the morphisms

B ⊗L

A C → B ⊗A C → C

and using arguments similar to the ones used in the proof of the implication (3)=⇒(4)in Theorem 6.3.

Remark 9.4. In the “expanded” form, Definition 9.1 means that for some (or, equiv-alently, for any) projective resolution 0 ← C ← P• in A-mod the complex 0 ← C ←B ⊗A P• is admissible (cf. Remark 6.4).

It follows from Theorem 6.7 (4) (see also Proposition 3.6 of [48]) that any homologicalepimorphism of augmented ⊗-algebras is a weak homological epimorphism.

Henceforth we shall use the notion of Hopf ⊗-algebra. Formally, a Hopf ⊗-algebra isdefined as a Hopf algebra in a symmetric monoidal category of complete LCS endowedwith the functor ⊗ of completed projective tensor product. For details, see [36, 5, 47, 48];we just remark that the definition of Hopf ⊗-algebra can be obtained from the usualalgebraic definition of Hopf algebra by replacing vector spaces with complete LCS, andthe symbol ⊗, with the symbol ⊗. We also remark that the Arens–Michael envelope ofa Hopf ⊗-algebra H has a canonical structure of a Hopf ⊗-algebra, which is uniquelydetermined by the requirement that ιH : H → H be a morphism of Hopf ⊗-algebras;see [48, 6.7].

The next result is Proposition 3.7 from [48].



Proposition 9.9. Suppose f : U → H is a morphism of Hopf ⊗-algebras with invertibleantipodes. Then f is a homological epimorphism if and only if f is a weak homologicalepimorphism.

Next, we recall the definition of the Koszul complex (see, for example, [6]). Suppose Ais a commutative algebra and x = (x1, . . . , xn) is a set of elements of A. For anyp = 0, . . . , n set Kp(x, A) = A⊗

∧pCn. Choose a basis e1, . . . , en in Cn and for any

p = 1, . . . , n define a morphism dp : Kp(x, A) → Kp−1(x, A) by the formula

dp(a ⊗ ei1 ∧ · · · ∧ eip) =

p∑k=1

(−1)k−1xika ⊗ ei1 ∧ · · · ∧ eik

∧ · · · ∧ eip.

The sequence of A-modules K•(x, A) = (Kp(x, A), dp)0≤p≤n forms a chain complex,called the Koszul complex of x ∈ An.

If A is an augmented commutative algebra and xi ∈ Ker εA for all i = 1, . . . , n, thenwe can also define the augmented Koszul complex 0 ← C

εA←−− K•(x, A).

Proposition 9.10. Let A be an augmented ⊗-algebra containing the polynomial algebraC[x1, . . . , xn] as a dense augmented subalgebra. Suppose that for any p = 0, . . . , n−1 thereexist continuous linear operators T

(p)1 , . . . , T

(p)n : A → A such that for any homogeneous

polynomial f of degree m we have

T(p)i (f) =

1p + m

∂f

∂xi(i = 1, . . . , n).

Then the embedding C[x1, . . . , xn] → A is a weak homological epimorphism.6

Proof. Set A0 = C[x1, . . . , xn], x = (x1, . . . , xn) and consider the augmented Koszulcomplex

(9.7) 0 ← Cε←− K•(x, A0).

It is well known (see, for example, [6]) that complex (9.7) is exact and is thus a projectiveresolution of C in A0-mod. Notice that since A0 is dense in A, the canonical morphismA ⊗A0 C → C is an isomorphism. Therefore, applying the functor A ⊗A0 ( · ) to (9.7), wehave an augmented Koszul complex

(9.8) 0 ← Cε←− K•(x, A).

Notice that (9.7) is a dense subcomplex of (9.8).For each p = 0, . . . , n−1 define an operator hp : Kp(x, A) → Kp+1(x, A) by the formula

hp(a ⊗ ω) =n∑

i=1

T(p)i (a) ⊗ ei ∧ ω (a ∈ A, ω ∈

∧pCn).

We also define an operator h−1 : C → A, 1C → 1A. It is known (see, for example, [31,18.7.6 (3)]) that the restrictions of the operators hp to the subcomplex (9.7) yield acontracting homotopy of the latter. As hp is continuous and (9.7) is dense in (9.8),it follows that hp is a contracting homotopy of the complex (9.8). Therefore, (9.8)is split in LCS, which means that A0 → A is a weak homological epimorphism (seeRemark 9.4).

Let g be a finite-dimensional Lie algebra and h ⊂ g an ideal of codimension 1. Fix somex ∈ g\h and set R = U(h). It is easy to see that there is an algebra isomorphism U(g) ∼=R[x; δ], where δ = adx (see the discussion preceding Example 5.1). Identifying U(g)

6Of course, C[x1, . . . , xn] is viewed here as a ⊗-algebra relative to the strongest locally convextopology.



with R[x; δ], and its Arens–Michael envelope U(g), with O(C, Rδ; δ) (see Theorem 5.1),we have Hopf ⊗-algebra structures on R[x; δ] and O(C, Rδ; δ); moreover, the canonicalmorphism ιR[x;δ] : R[x; δ] → O(C, Rδ; δ) becomes a morphism of Hopf ⊗-algebras (see thediscussion preceding Proposition 9.9).

We shall consider Rδ as the closed subalgebra of H = O(C, Rδ; δ) consisting of con-stant functions. Clearly, Rδ is complemented in H as a subspace; therefore Rδ ⊗Rδ iscanonically identified with a closed subalgebra of H ⊗H. Let ∆H and SH denote, re-spectively, the comultiplication and the antipode of H. Notice that ∆H(Rδ) ⊂ Rδ ⊗Rδ

and SH(Rδ) ⊂ Rδ (this immediately follows from the facts that R is a Hopf subalgebraof R[x; δ], ι : R[x; δ] → H is a morphism of Hopf ⊗-algebras, and ι(R) = Rδ). Thus Rδ

is a Hopf ⊗-algebra and the canonical homomorphism j = ι|R : R → Rδ is a morphismof Hopf ⊗-algebras. Notice also that S2

H = 1H immediately implies that Rδ is a Hopf⊗-algebra with an invertible antipode.

Corollary 9.11. Let g be a finite-dimensional Lie algebra, h ⊂ g a codimension 1 ideal,and R = U(h). Fix some x ∈ g\h and set δ = adx : R → R. Suppose that j : R → Rδ is aweak homological epimorphism. Then ιU(g) : U(g) → U(g) is a homological epimorphism.

Proof. Taking account of Proposition 9.9 and the above discussion, this is a particularcase of Corollary 9.8.

Theorem 9.12. Suppose A is any of the following algebras:

(i) A = Oalgq (Cn), the multiparameter quantum affine space (where q = (qij)n

i,j=1 isa multiplicatively antisymmetric (n × n)-matrix such that either |qij | ≥ 1 for alli < j or |qij | ≤ 1 for all i < j);

(ii) A = Oalgq

((C×)n

), the multiparameter quantum torus (where q = (qij)n

i,j=1 is amultiplicatively antisymmetric (n × n)-matrix);

(iii) A is an algebra satisfying the conditions of Theorem 5.18;(iv) A = A1(q), the quantum Weyl algebra (where q ∈ C×);(v) A = Mq(2), the algebra of quantum (2 × 2)-matrices (where q ∈ C×);(vi) A = U(g), where g is a Lie algebra with basis x, y and commutator [x, y] = y;(vii) A = U(g), where g is the three-dimensional Heisenberg Lie algebra.

Then ιA : A → A is a homological epimorphism.

Proof. (i) Notice first that we have an obvious isomorphism

Oalgq (Cn) ∼= O

algq (Cn),

where qij = qn−i,n−j , sending xi to xn−i (cf. the proof of Proposition 5.12 (ii)). Therefore,without loss of generality we may assume that |qij | ≥ 1 for all i < j.

We now induct on n. When n = 0 the assertion is obvious. Suppose that it is true formatrices of order n− 1. Set q = (qij)n−1

i,j=1 and consider the algebra R = Oalgq

(Cn−1). By

Corollary 5.13 (i), for its Arens–Michael envelope R = Oq(Cn−1), representation (5.11)holds (with n replaced by n−1). As we said before, there is an isomorphism A = R[xn; α],where the endomorphism α : R → R is uniquely determined by the conditions α(xi) =qnixi (i = 1, . . . , n − 1). Since |qni| ≤ 1, it follows from (5.11) that ‖α(a)‖ρ ≤ ‖a‖ρ forany a ∈ R and any ρ > 0. Therefore, α is m-localizable, and to complete the proof,apply Corollary 9.6.

(ii) If |qij | = 1 at least for one pair i, j, then A = 0 (see Proposition 5.23), and thereis nothing to prove. If |qij | = 1 for all i, j, then, using Corollaries 5.22 and 9.7, we canargue as in the proof of (i).



(iii) Set R = Oalgq (Cn); then A = R[xn+1, σ, δ], where the endomorphism σ : R → R

and the σ-derivation δ : R → R are uniquely determined by the conditions

σ(xi) = λixi (1 ≤ i ≤ n), δ(x1) = µxγ , δ(xk) = 0 (k ≥ 2)

(see the proof of Theorem 5.18). Furthermore, the canonical extension of σ and δ to R

form an m-localizable family σ, δ (ibid.). Now use (i) and Corollary 9.6.(iv) If q = 1, then A = 0 (see Example 3.4), and there is nothing to prove. If q = 1,

then A satisfies the conditions of Theorem 5.18 (see Example 5.5), and everything reducesto (iii).

(v) If q = ±1, then A satisfies the conditions of Theorem 5.18 (see Example 5.6) andthe desired assertion follows from (iii). If q = −1, then A = Oq′(C4) (see (5.30)) andeverything reduces to (i). Finally, when q = 1 we have A = C[a, b, c, d]. Now applyProposition 4.3 from [67] (or use (i) with the matrix q consisting of ones).

(vi) Set h = Cy, R = U(h) = C[y], and δ = adx : R → R. Then Rδ = C[[y]] (seeExample 5.1). Consider the linear operator

T(0)1 : C[[y]] → C[[y]],

∑n

cnyn →∑n≥1

cnyn−1.

Clearly, T(0)1 is continuous and satisfies the conditions of Proposition 9.10. There-

fore, j : R → Rδ is a weak homological epimorphism, and the desired assertion followsfrom Corollary 9.11.

(vii) Set h = spany, z, R = U(h) = C[y, z], and δ = adx : R → R. Then

Rδ =

a =∞∑

i,j=0

cijyizj : ‖a‖ρ =

∞∑i,j=0

|cij |i!

(i + j)!ρi+j < ∞ ∀ρ > 0

(see Example 5.2). For p = 0, 1 consider the linear operators

T(p)1 : Rδ → Rδ,

∑i,j≥0

cijyizj →

∑i≥1, j≥0

ciji

i + j + pyi−1zj ,

T(p)2 : Rδ → Rδ,

∑i,j≥0

cijyizj →

∑i≥0, j≥1

cijj

i + j + pyizj−1.

It is not difficult to check that these operators are well-defined and continuous. Indeed,for any a =

∑i,j cijy

izj ∈ Rδ and any ρ > 0 we have∑i≥1, j≥0

|cij |i

i + j + p

(i − 1)!(i + j − 1)!

ρi+j−1 ≤∑

i≥1, j≥0

|cij |i!

(i + j)!ρi+j−1 ≤ ρ−1‖a‖ρ,

whence ‖T (p)1 (a)‖ρ ≤ ρ−1‖a‖ρ. Furthermore,∑

i≥0, j≥1

|cij |j

i + j + p

i!(i + j − 1)!

ρi+j−1 ≤∑

i≥0, j≥1

|cij |2j−1i!(i + j)!

ρi+j−1

≤∑

i≥0, j≥1

|cij |i!

(i + j)!(2ρ)i+j−1 ≤ (2ρ)−1‖a‖2ρ,

and therefore∥∥T (p)

2 (a)∥∥

ρ≤ (2ρ)−1‖a‖2ρ. Thus the operators T

(p)1 and T

(p)2 are well-

defined, continuous, and, clearly, satisfy the conditions of Proposition 9.10. Thereforej : R → Rδ is a weak homological epimorphism, and the desired assertion follows fromCorollary 9.11.



10. Appendix: Derived categories

For the convenience of the reader, in this appendix we mention some facts from thetheory of derived categories. Notice that most of the standard references [23, 18, 30]only consider the derived categories of abelian categories, whereas the typical categoriesof functional analysis are not abelian. We shall consider the more general case of exactcategories in the sense of Quillen [56]. We shall mostly follow the survey paper by B.Keller [32].

Let A be an additive category. We shall always assume that idempotents split in A;i.e., each morphism p : X → X in A such that p2 = p has a kernel. An exact pairin A is any sequence X

i−→ Yp−→ Z, such that i = ker p and p = coker i. Suppose E

is some class of exact pairs in A, closed under isomorphisms. Elements of E will becalled admissible pairs. A morphism i : X → Y (resp., p : Y → Z) is said to be anadmissible monomorphism (resp., admissible epimorphism) if it is part of an admissiblepair X

i−→ Yp−→ Z.

An exact category is an additive category A endowed with a class of exact pairs E

satisfying the following axioms:

Ex0. The identity morphism of a zero object is an admissible epimorphism.Ex1. The composition of two admissible epimorphisms is an admissible epimor-

phism.Ex1. The composition of two admissible monomorphisms is an admissible mono-

morphism.Ex2. Each diagram

Y ′

X

p Y

where p is an admissible epimorphism, can be extended to a pull-back square

X ′

p′ Y ′

X

p Y

where p′ is an admissible epimorphism.Ex2. Each diagram

X

i Y

X ′

where i is an admissible monomorphism, can be extended to a push-outsquare

X

i Y

X ′ i′ Y ′

where i′ is an admissible monomorphism.



If A is an exact category, then the dual category A is also exact, provided theadmissible pairs in A are defined as the pairs X

i−→ Yp−→ Z such that the pair Z

p−→Y

i−→ X in A is admissible.

Example 10.1. Each additive category A can be made exact if E is defined as the classof all split exact sequences. The corresponding exact category will be denoted Aspl. Weshall then say that A is endowed with the split exact structure.

Example 10.2 (quasi-abelian and abelian categories). Suppose that A is an additivecategory with kernels and cokernels, and let E denote the class of all exact pairs in A. If,under these assumptions, A satisfies axioms Ex2 and Ex2, then A is said to be quasi-abelian [53, 61] (or, in the original terminology, semi-abelian [58]). Since axioms Ex0,Ex1, and Ex1 are automatically satisfied, any quasi-abelian category endowed with aclass of all exact pairs becomes an exact category. Notice that many standard categoriesof functional analysis are quasi-abelian [54]. The examples include: the category LCS oflocally convex spaces (LCS), its full subcategory consisting of Hausdorff LCS, the cate-gory of Frechet spaces, the categories of all normed spaces and of all Banach spaces, andothers. Notice, however, that the category of all complete LCS is not quasi-abelian [54].

Recall that A is said to be abelian if any monomorphism in it is a kernel, and anyepimorphism is a cokernel (see, for example, [39]). Any abelian category is quasi-abelianand therefore is an exact category.

Example 10.3. Suppose A is an additive category with kernels and cokernels, B is anexact category, and : A → B is an additive functor preserving kernels and cokernels.Then it is easy to check that A becomes an exact category if one defines admissible pairsin A as the pairs whose images under are admissible in B. An important particularcase of this construction is when B = Bspl. In this case we denote the exact categoryobtained by AB.

For a given additive category A the category of all chain complexes in A is denotedC(A). Its full subcategory consisting of complexes bounded on the right (resp., boundedon the left, bounded), i.e., complexes X ∈ Ob (C(A)) such that Xn = 0 for all n > N(resp., for all n < N, for all |n| > N) holds for some N ∈ Z, is denoted C−(A) (resp.,C+(A), Cb(A)). The differential in a chain complex X mapping Xn to Xn+1 will bedenoted dn

X . Recall that the homotopy category H(A) is defined as follows: it hasthe same objects as C(A), and the group of morphisms Hom H(A)(X, Y ) is defined asthe quotient of Hom C(A)(X, Y ) by the subgroup of all null-homotopic morphisms. Thecategories H−(A), H+(A), and Hb(A) are defined similarly.

Following the convention, we identify chain complexes with cochain complexes byviewing a chain complex (Xn, dn) as a cochain complex (Xn, dn), where Xn = X−n anddn = d−n. Notice also that there is an isomorphism C(A) ∼= C(A), assigning to acomplex (Xn, dn) in A a complex (Xn, dn) in A, such that Xn = X−n and dn = d−n−1.This isomorphism, in turn, gives rise to isomorphisms C−(A) ∼= C+(A) and H−(A) ∼=H+(A).

For any cochain complex X in A, let X[1] be the complex such that X[1]n = Xn+1

and dnX[1] = −dn+1

X for all n ∈ Z. If f : X → Y is a morphism in C(A), then themorphism f [1] : X[1] → Y [1] is defined by the formula f [1]n = fn+1. Thus we havea functor X → X[1] from C(A) in C(A), called the suspension functor (or the shiftfunctor). Similarly, one defines the suspension functor from H(A) to H(A).



Recall that for any morphism f : X → Y in C(A) its cone M(f) is the complex definedas follows:

M(f)n = Xn+1 ⊕ Y n, dnM(f) =

(dn

X[1] 0fn dn

Y

).

Each morphism f : X → Y is part of a sequence

(10.1) Xf−→ Y

α(f)−−−→ M(f)β(f)−−−→ X[1],

where α(f) : y → (0, y) and β(f) : (x, y) → x.A triangle in H(A) is any sequence of morphisms of the form X → Y → Z → X[1]. A

morphism from this triangle to a triangle X ′ → Y ′ → Z ′ → X ′[1] is a triple of morphisms(u, v, w) which is part of a commutative diagram

X

u

Y

v

Z

w

X[1]

u[1]

X ′ Y ′ Z ′ X ′[1]

A triangle in H(A) is said to be distinguished if it is isomorphic to a triangle of theform (10.1).

The category H(A) endowed with the functor X → X[1] and the class of all distin-guished triangles is a triangulated category. We do not give here the precise definition ofa triangulated category (see, for example, [18] or [30]); we just remark that a triangulatedcategory (C, T ) is an additive category C endowed with an automorphism T : C → C anda class of sequences X → Y → Z → T (X), called distinguished triangles, satisfyingcertain axioms.

For the proof of the following standard fact, see, for example, [18] or [30].

Proposition 10.1. Suppose (u, v, w) is a morphism of distinguished triangles in a trian-gulated category (C, T ) such that two of the morphisms u, v, w are isomorphisms. Thenso is the third.

We continue to assume that A is an exact category. A complex X in A is said to beadmissible if for any n ∈ Z the differential dn has a kernel, and the canonical sequenceKer dn → Xn → Ker dn+1 is an admissible pair. A morphism f : X → Y in C(A) iscalled a quasi-isomorphism if its cone M(f) is admissible.

Example 10.4. If A is abelian, then a complex in A is admissible if and only if it isexact, and a morphism f : X → Y in C(A) is a quasi-isomorphism if and only if it inducesan isomorphism in cohomology.

Example 10.5. Suppose that A is an additive category with kernels and cokernels.Recall that a morphism f : X → Y is said to be strict if the induced morphism Coim f →Im f is an isomorphism. Notice that A is abelian if and only if each morphism in it isstrict. If A is quasi-abelian (see Example 10.2), then a complex in A is admissibleif and only if it is strictly exact, i.e., when it is exact and each differential dn is astrict morphism [53, 61]. In particular, a complex in the quasi-abelian category LCS isadmissible if and only if it is topologically exact [25], i.e., when each dn is an open mapfrom Xn to Ker dn+1 [54].

Example 10.6. For an arbitrary additive category B a complex X ∈ Ob (C(B)) isadmissible in the exact category Bspl if and only if it is split (i.e., null-homotopic). Inthis case quasi-isomorphisms are precisely homotopy equivalences.



Example 10.7. As an immediate consequence of the preceding example, we have thatin the exact category AB (see Example 10.3) a complex X is admissible if and only ifX is split in B, and a morphism of complexes f : X → Y is a quasi-isomorphism if andonly if f : X → Y is a homotopy equivalence.

Suppose (C, T ) and (C′, T ′) are triangulated categories. A triangular functor from(C, T ) to (C′, T ′) is a pair (F, α), where F : C → C′ is an additive functor and α : FT →T ′F is a natural isomorphism such that for any distinguished triangle X

f−→ Yg−→ Z

h−→T (X) in C the triangle

FXFf−−→ FY

Fg−−→ FZαF T (X)Fh−−−−−−−→ T ′(FX)

is distinguished in C′. A morphism of triangular functors and a triangular equivalence oftriangulated categories are defined in an obvious way.

The derived category of an exact category A is defined as the localization of thehomotopy category H(A) by the system of all quasi-isomorphisms. In other words,the derived category is a pair (D(A), QA) consisting of a category D(A) and a functorQA : H(A) → D(A) sending quasi-isomorphisms to isomorphisms; also, for any category C

and any functor F : H(A) → C sending quasi-isomorphisms to isomorphisms there existsa unique functor G : D(A) → C such that F = G QA. Moreover, D(A) has a canonicalstructure of a triangulated category and the functor QA is triangular. The derived cat-egory always exists and is unique up to a triangular equivalence. If C is a triangulatedcategory and F : H(A) → C is a triangular functor sending admissible objects to zero,then it sends quasi-isomorphisms to isomorphisms, and the functor G : D(A) → C, whoseexistence is postulated above, is also triangular.

In a similar way (replacing H(A) by H−(A), H+(A), or Hb(A)) one defines derivedcategories D−(A), D+(A), and Db(A). Notice that there are isomorphisms D−(A) ∼=D+(A) and D+(A) ∼= D−(A). Moreover, D−(A), D+(A), and Db(A) can be identifiedwith full subcategories of D(A). For more details, see [32] or (for the abelian case) [18, 30].

Associating with each X ∈ Ob (A) the complex whose degree zero term is X andall other terms are zero, we have a fully faithful additive functor A → C(A). Its com-positions with the canonical functors C(A) → H(A) and C(A) → D(A) are also fullyfaithful functors; for this reason A is customarily identified with the full subcategoriesof C(A), H(A), and D(A) consisting of complexes concentrated in degree zero. A similarconvention is used for C∗(A), H∗(A), and D∗(A), where ∗ ∈ +,−, b.

Let A, B be exact categories and F : A → B an additive covariant functor. In anobvious way, F gives rise to a triangular functor H−(F ) : H−(A) → H−(B). The leftderived functor of F is defined as a pair (LF, s), consisting of a triangular functor

LF : D−(A) → D−(B)

and a morphism of triangular functors

(10.2) sF : LF QA → QB H−(F ),

such that for any other such pair (G, s) there exists a unique morphism of triangularfunctors α : G → LF with s = sF (αQA).

Right derived functors, as well as derived functors of contravariant functors can bedefined by passing to the dual categories. For each functor F : A → B we have thefunctors F : A → B, F : A → B, and F

: A → B, acting the same way as F ;moreover, if F is covariant (resp., contravariant), then so is F

, whereas F and F arecontravariant (resp., covariant).



If F : A → B is an additive covariant functor, then its right derived functor is definedas the functor RF = L(F

) : D+(A) → D+(B). If F is contravariant, then we setRF = L(F) : D−(A) → D+(B) and LF = L(F ) : D+(A) → D−(B).

A functor F : A → B is said to be exact if it sends admissible complexes to admissibleones. If F is exact, then, as follows from the definition of the derived category, thereexists a unique triangular functor F ′ : D(A) → D(B) such that F ′ QA = QB H(F ). Itis easy to see that the restriction of F ′ to D−(A) (resp., D+(A)) is the left (resp., right)derived functor of F ; moreover, (10.2) is the identity morphism.

Recall that an object P ∈ Ob (A) is said to be projective if for any admissible epimor-phism X → Y in A the induced map Hom A(P, X) → Hom A(P, Y ) is surjective. Onesays that A has enough projectives if for any X ∈ Ob (A) there are a projective objectP ∈ Ob (A) and an admissible epimorphism P → X.

Example 10.8. Under the assumptions of Example 10.3, assume in addition that thefunctor : A → B has a left adjoint F : B → A and that B = Bspl. Then for anyadmissible epimorphism Y → Z in AB the morphism Y → Z is a retraction in B,and therefore for any E ∈ Ob (B) the map Hom B(E, Y ) → Hom B(E, Z) is surjective.Since F and are adjoint, we conclude that F (E) is projective in AB. Furthermore, sincethe composition → F → is the identity morphism of (see, for example, [39,4.1.1]), we have that for any X ∈ Ob (A) the morphism F (X) → X is a retractionin B, or, equivalently, that F (X) → X is an admissible epimorphism. Therefore, AB

has enough projectives. Moreover, an object X is projective if and only if the canonicalmorphism F (X) → X is a retraction.

Suppose that A is an exact category with enough projectives. Then for any complexX ∈ Ob (C−(A)) there are a complex P ∈ Ob (C−(A)) consisting of projectives and aquasi-isomorphism P → X. Any two complexes P, P ′ with this property are homotopyequivalent (and therefore quasi-isomorphic). Any such complex P together with a quasi-isomorphism P → X is called a projective resolution of X. Let P be the full subcategoryof A consisting of projectives. Then, as follows from the above discussion, the canonicalfunctor D−(P) → D−(A) is an equivalence of triangulated categories. On the other hand,any acyclic complex consisting of projectives is homotopy equivalent to a zero complex;hence QP : H−(P) → D−(P) is an equivalence. The left derived functor LF : D−(A) →D−(B) can now be defined as the composition of the functors

D−(A) ∼←− D−(P) ∼←− H−(P) → H−(A)H−(F )−−−−→ H−(B)

QB−−→ D−(B).

Notice that having enough projectives is a rather strict sufficient condition for the ex-istence of a left derived functor. For much more general conditions, see [32] or (in theabelian case) [18, 30].

An object I ∈ Ob (A) is said to be injective if it is projective as an object of A. Asfollows from the above, for the existence of the right derived functor of a covariant functoror of the left derived functor of a contravariant functor it suffices that A has enoughinjectives, i.e., for any X ∈ Ob (A) there is an admissible monomorphism X → I, where Iis injective. Similarly, for the existence of the right derived functor of a contravariantfunctor it suffices that A has enough projectives.

If the category B is quasi-abelian, then for any n ∈ Z the cohomology functorHn : C−(B) → B sends quasi-isomorphisms to isomorphisms. This is proved the sameway as for abelian categories, using the long cohomology exact sequence (see [34]).7

Therefore, the functor Hn extends to a functor D−(B) → B, also denoted Hn. The

7In fact, a simple argument shows that this is true in any exact category B with kernels and cokernels.We skip the proof, since we do not need such a level of generality.



composition H−n LF : D−(A) → B (and its restriction to A) is called the n-th classicalleft derived functor of F and is denoted LnF . Similarly, the n-th classical right derivedfunctor RnF : D+(A) → B is defined as the composition Hn RF .

Example 10.9. Suppose that A has enough projectives. Fix some Y ∈ Ob (A) andconsider the functor

Hom A( · , Y ) : A → Ab,

where Ab is the (abelian) category of abelian groups. Its right derived functor is denotedRHom A( · , Y ), and its n-th classical right derived functor is denoted Ext n

A( · , Y ). Thefunctor RHom A( · , Y ) extends to a bifunctor

RHom A : D−(A) × D+(A) → D+(Ab)

(the proof is the same as in the case of abelian categories; see, for example, [23, I.6]or [30, 1.10]). Notice that we have a natural isomorphism

Ext nA(X, Y ) ∼= Hom D(A)(X, Y [n])

(ibid. or [18, III.6.15]).

Suppose A, B, and C are exact categories and F : A → B and G : B → C are additivecovariant functors. Suppose that F , G, and GF have left derived functors. It follows fromthe definition of the derived functor that there is a morphism of functors c : LG LF →L(GF ) making the following diagram commute:

LG LF QA(LG)sF

cQA

LG QB H−(F )

sG(H−(F ))

L(GF ) QA sGF

QC H−(G) H−(F )

If c is an isomorphism, then we say that the theorem on the derived functor of thecomposition holds for F and G. For some of the sufficient conditions guaranteeing this,see [32] or (in the abelian case) [18, 30]. In particular, this theorem holds if A and B

have enough projectives, and F sends projectives to projectives.Suppose A and B are exact categories and F : A → B is an additive covariant functor

having a derived functor. As before, we identify A and B with the full subcategoriesof D−(A) and, respectively, D−(B), and consider F and LF as functors from A to D−(B).Then the morphism (10.2) gives rise to a morphism of functors LF → F . An objectX ∈ Ob (A) is said to be (left) F -acyclic if LF (X) → F (X) is an isomorphism in D−(B).

Example 10.10. A projective object is acyclic relative to any functor (see [32]).

Example 10.11. If F is exact, then any object of A is F -acyclic.

Example 10.12. Suppose A is a ⊗-algebra and Y is a left A-⊗-module. Let Vectdenote the abelian category of vector spaces. Then Y is flat (in the sense of [27, 25]) ifand only if Y is acyclic relative to the functor X ⊗A ( · ) : A-mod → Vect for any rightA-⊗-module X.

The author is grateful to A. Ya. Helemskii for his attention to this work.

References

[1] V. A. Artamonov, Algebras of quantum polynomials, Itogi Nauki Tekh. Ser. Sovrem. Mat. Prilozh.Temat. Obz. vol. 26, pp. 5–34, VINITI. Moscow, 2002. (Russian)

[2] W. B. Arveson, Subalgebras of C∗-algebras, Acta Math. 123 (1969), 141–224. MR0253059 (40:6274)[3] W. B. Arveson, Subalgebras of C∗-algebras, II, Acta Math. 128 (1972), no. 3–4, 271–308.

MR0394232 (52:15035)


http://www.ams.org/mathscinet-getitem?mr=0253059





[4] B. Blackadar and J. Cuntz, Differential Banach algebra norms and smooth subalgebras of C∗-algebras , J. Operator Theory 26 (1991), no. 2, 255–282. MR1225517 (94f:46094)

[5] P. Bonneau, M. Flato, M. Gerstenhaber, and G. Pinczon, The hidden group structure of quantumgroups: strong duality, rigidity and preferred deformations, Comm. Math. Phys. 161 (1994), 125–156. MR1266072 (95b:17016)

[6] N. Bourbaki, Elements de mathematique, Algebre, Chapitre 10, Algebre homologique, Masson,Paris, 1980. MR610795 (82j:18022)

[7] J. Cuntz and D. Quillen, Algebra extensions and nonsingularity, J. Amer. Math. Soc. 8 (1995),251–289. MR1303029 (96c:19002)

[8] J. Cuntz, Bivariante K-Theorie fur lokalkonvexe Algebren und der Chern-Connes-Charakter, Doc.Math. 2 (1997), 139–182. MR1456322 (98h:19006)

[9] J. Cuntz, Cyclic theory and the bivariant Chern–Connes character, Noncommutative geometry,Lecture Notes in Math., vol. 1831, pp. 73–135, Springer, Berlin, 2004. MR2058473 (2005e:58007)

[10] H. G. Dales, Banach algebras and automatic continuity, London Mathematical Society Monographs,New Series, vol. 24, Oxford Science Publications, The Clarendon Press, Oxford University Press,New York, 2000. MR1816726 (2002e:46001)

[11] K. R. Davidson, Free semigroup algebras, A survey, Systems, approximation, singular integral oper-ators, and related topics (Bordeaux, 2000). Oper. Theory Adv. Appl. 129, pp. 209–240, Birkhauser,Basel, 2001. MR1882697 (2003d:47104)

[12] K. R. Davidson and D. R. Pitts, The algebraic structure of non-commutative analytic Toeplitzalgebras, Math. Ann. 311 (1998), 275–303. MR1625750 (2001c:47082)

[13] K. R. Davidson and G. Popescu, Noncommutative disc algebras for semigroups, Canad. J. Math.

50 (1998), 290–311. MR1618326 (99f:46105)[14] E. E. Demidov, Modules over a quantum Weyl algebra, Vestnik Moskov. Univ. Ser. I Mat. Mekh.

1993, no. 1, 53–56, MR1293939 (95e:17015)[15] A. A. Dosiev, Algebras of holomorphic functions in elements of a nilpotent Lie algbera and Taylor

localizations, Preprint.[16] A. A. Dosiev, Homological dimensions of the algebra of entire functions of elements of a nilpotent

Lie algebra, Funktsional. Anal. i Prilozhen. 37 (1) (2003), 73–77. MR1988010 (2004d:46054)[17] W. Geigle and H. Lenzing, Perpendicular categories with applications to representations and

sheaves, J. Algebra 144 (1991), 273–343. MR1140607 (93b:16011)[18] S. I. Gelfand and Y. I. Manin, Methods of homological algebra, Springer Monographs in Mathe-

matics, second edition, Springer-Verlag, Berlin, 2003. MR1950475 (2003m:18001)[19] M. Gerstenhaber and S. D. Schack, Algebraic cohomology and deformation theory, Deformation the-

ory of algebras and structures and applications (Il Ciocco, 1986). NATO Adv. Sci. Inst. Ser. C Math.Phys. Sci., vol. 247, pp. 11–264, Kluwer Acad. Publ., Dordrecht, 1988. MR981619 (90c:16016)

[20] K. R. Goodearl, Quantized coordinate rings and related Noetherian algebras, Proceedings of the35th Symposium on Ring Theory and Representation Theory (Okayama, 2002), pp. 19–45, Symp.Ring Theory Represent. Theory Organ. Comm., Okayama, 2003. MR1969455

[21] K. R. Goodearl and E. S. Letzter, Quantum n-space as a quotient of classical n-space, Trans. Amer.Math. Soc. 352 (2000), 5855–5876. MR1781280 (2001e:16061)

[22] R. S. Hamilton, The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. (N. S.) 7(1982), 65–222. MR656198 (83j:58014)

[23] R. Hartshorne, Residues and duality, Lecture notes of a seminar on the work of A. Grothendieck,given at Harvard 1963/64. With an appendix by P. Deligne, Lecture Notes in Mathematics, vol. 20,

Springer-Verlag, Berlin, 1966. MR0222093 (36:5145)[24] A. Ya. Helemskii, Banach and locally convex algebras, Oxford Science Publications, The Clarendon

Press Oxford University Press, New York, 1993. MR1231796 (94f:46001)[25] A. Ya. Helemskii, The homology of Banach and topological algebras, Mathematics and its Appli-

cations (Soviet Series), vol. 41, Kluwer Academic Publishers Group, Dordrecht, 1989. MR1093462(92d:46178)

[26] A. Ya. Helemskii, Homological methods in the holomorphic calculus of several operators in Banachspace, after Taylor, Uspekhi Mat. Nauk 36 (1981), 127–172. MR608943 (82e:47020)

[27] A. Ya. Helemskii, A certain class of flat Banach modules and its applications. Vestnik Moskov. Univ.Ser. I Mat. Meh. 27 (1972), no. 1, 29–36. MR0295080 (45:4148)

[28] G. Hochschild, Relative homological algebra, Trans. Amer. Math. Soc. 82 (1956), 246–269.MR0080654 (18:278a)

[29] D. A. Jordan, A simple localization of the quantized Weyl algebra, J. Algebra 174, (1995), 267–281.MR1332871 (96m:16035)




















































[30] M. Kashiwara and P. Schapira, Sheaves on manifolds, Grundlehren der Mathematischen Wis-senschaften, vol. 292, Corrected reprint of the 1990 original, Springer-Verlag, Berlin, 1994.MR1074006 (92a:58132)

[31] C. Kassel, Quantum groups, Graduate Texts in Mathematics, vol. 155, Springer-Verlag, New York,1995. MR1321145 (96e:17041)

[32] B. Keller, Derived categories and their uses, Handbook of algebra, M. Hazewinkel, ed., Amsterdam,North-Holland, 1996, vol. 1, pp. 671–701. MR1421815 (98h:18013)

[33] E. Kissin and V. S. Shul’man, Differential properties of some dense subalgebras of C∗-algebras,Proc. Edinburgh Math. Soc. (2) 37 (1994), 399–422. MR1297311 (95j:46088)

[34] Y. A. Kopylov and V. I. Kuz’minov, Exactness of the cohomology sequence for a short exact sequenceof complexes in a semiabelian category, Siberian Adv. Math. 13 (3) (2003), 72–80. MR2028407

[35] Lectures on q-analogues of Cartan domains and associated Harish-Chandra modules, L. Vaksman(ed.), Kharkov, 2001. Preprint arXiv:math.QA/0109198.

[36] G. L. Litvinov, On double topological algebras and Hopf topological algebras, Trudy Sem. Vektor.Tenzor. Anal. 18 (1978), 372–375; English transl., Selecta Math. Soviet. 10 (4) (1991), 339–343.MR0504538 (80a:46024)

[37] D. Luminet, A functional calculus for Banach PI-algebras, Pacific J. Math. 125 (1986), 127–160.MR860755 (88c:46060)

[38] D. Luminet, Functions of several matrices, Boll. Un. Mat. Ital. B. (7) 11 (1997), 563–586.MR1479512 (98i:32007)

[39] S. Mac Lane, Categories for the working mathematician, Graduate Texts in Mathematics, vol. 5,second edition, Springer-Verlag, New York, 1998. MR1712872 (2001j:18001)

[40] Yu. I. Manin, Some remarks on Koszul algebras and quantum groups, Ann. Inst. Fourier (Grenoble)37 (1987), 191–205. MR927397 (89e:16022)

[41] J. C. McConnell and J. J. Pettit, Crossed products and multiplicative analogues of Weyl algebras,J. London Math. Soc. (2) 38 (1988), 47–55. MR949080 (90c:16011)

[42] A. I. Markushevich, Theory of functions of a complex variable, vols. I, II, III, Chelsea PublishingCo., New York, 1977. MR0444912 (56:3258)

[43] R. Meyer, Analytic cyclic cohomology, Ph.D. Thesis, Westfalische Wilhelms-Universitat Munster,1999. Preprint arXiv.org:math.KT/9906205.

[44] R. Meyer, Embeddings of derived categories of bornological modules, PreprintarXiv.org:math.FA/0410596.

[45] B. Mitiagin, S. Rolewicz, and W. Zelazko W., Entire functions in B0-algebras, Studia Math. 21(1961/1962), 291–306. MR0144222 (26:1769)

[46] A. Neeman and A. Ranicki, Noncommutative localization and chain complexes, I. Algebraic K- andL-theory, Preprint arXiv.org:math.RA/0109118.

[47] M. Pflaum and M. Schottenloher, Holomorphic deformations of Hopf algebras and applications toquantum groups, J. Geometry and Physics 28 (1988), 31–44. MR1653122 (99k:58018)

[48] A. Yu. Pirkovskii, Stably flat completions of universal enveloping algebras, Dissertationes Math.(Rozprawy Math.) 441 (2006), 1–60. MR2283621 (2007k:46131)

[49] A. Yu. Pirkovskii, Arens–Michael enveloping algebras and analytic smash products, Proc. Amer.Math. Soc. 134 (2006), 2621–2631. MR2213741 (2007b:46135)

[50] G. Popescu, Non-commutative disc algebras and their representations, Proc. Amer. Math. Soc. 124(1996), 2137–2148. MR1343719 (96k:47077)

[51] G. Popescu, Noncommutative joint dilations and free product operator algebras, Pacific J. Math.186 (1998), 111–140. MR1665059 (2000k:46082)

[52] D. Proskurin and Yu. Samoılenko, Stability of the C∗-algebra associated with twisted CCR, Algebr.Represent. Theory 5 (2002), 433–444. MR1930972 (2003i:46059)

[53] F. Prosmans, Algebre homologique quasi-abelienne, Mem. DEA, Univ. Paris 13, Juin 1995.[54] F. Prosmans, Derived categories for Functional Analysis, Publ. Res. Inst. Math. Sci. 36 (2000),

19–83. MR1749013 (2001g:46156)[55] W. Pusz and S. L. Woronowicz, Twisted second quantization, Rep. Math. Phys. 27 (1989), 231–257.

MR1067498 (93c:81082)[56] D. Quillen, Higher Algebraic K-theory I, Lecture Notes in Math., vol. 341, pp. 85–147, Berlin,

Springer, 1973. MR0338129 (49:2895)[57] Y. V. Radyno, Linear equations and bornology, Beloruss. Gos. Univ., Minsk, 1982. (Russian)

MR685429 (85d:46006)[58] D. A. Raıkov, Semiabelian categories, Dokl. Akad. Nauk SSSR 188 (1969), 1006–1009. MR0255639

(41:299)


















































[59] W. Rudin, Functional analysis, International Series in Pure and Applied Mathematics, secondedition, McGraw-Hill, New York, 1991. MR1157815 (92k:46001)

[60] H. H. Schaefer. Topological vector spaces, Third printing corrected, Graduate Texts in Mathematics,vol. 3, Springer-Verlag, New York, 1971. MR0342978 (49:7722)

[61] J.-P. Schneiders, Quasi-abelian categories and sheaves, Mem. Soc. Math. Fr. (N. S.) No. 76 (1999).MR1779315 (2001i:18023)

[62] J.-P. Serre, Geometrie algebrique et geometrie analytique, Ann. Inst. Fourier (Grenoble) 6 (1955–1956), 1–42. MR0082175 (18:511a)

[63] S. Sinel’shchikov and L. Vaksman, On q-analogues of bounded symmetric domains and Dolbeaultcomplexes, Math. Phys. Anal. Geom. 1 (1998), 75–100. MR1687517 (2000f:58016)

[64] B. Stenstrom, Rings of quotients, Die Grundlehren der Mathematischen Wissenschaften, Band 217.Springer-Verlag, New York, 1975. MR0389953 (52:10782)

[65] M. E. Sweedler, Hopf algebras, Benjamin, New York, 1969. MR0252485 (40:5705)

[66] J. L. Taylor, Homology and cohomology for topological algebras, Adv. Math. 9 (1972), 137–182.MR0328624 (48:6966)

[67] J. L. Taylor, A general framework for a multi-operator functional calculus, Adv. Math. 9 (1972),183–252. MR0328625 (48:6967)

[68] J. L. Taylor, Functions of several noncommuting variables, Bull. Amer. Math. Soc. 79 (1973), 1–34.MR0315446 (47:3995)

[69] J. L. Taylor, Several complex variables with connections to algebraic geometry and Lie groups,Graduate Studies in Mathematics, vol. 46. American Mathematical Society, Providence, RI, 2002.MR1900941 (2004b:32001)

[70] L. Vaksman, The maximum principle for “holomorphic functions” in the quantum ball, Mat. Fiz.Anal. Geom. 10 (1) (2003), 12–28. MR1937043 (2004h:46088)

[71] L. L. Vaksman and D. L. Shklyarov, Integral representations of functions in the quantum disk I,Mat. Fiz. Anal. Geom. 4 (3) (1997), 286–308. MR1615820 (99g:81098)

[72] L. Waelbroeck, Topological vector spaces and algebras, Lecture Notes in Mathematics, vol. 230,Springer-Verlag, Berlin, 1971. MR0467234 (57:7098)

[73] W. Zelazko, Operator algebras on locally convex spaces, Proc. of the 5th International Conferenceon Topological Algebras and Applications, Athens, 2005, Contemporary Math. 427 (2007), 431–442.MR2326378

Russian Peoples’ Friendship University, 117198 Moscow, Russia

E-mail address: [email protected]; [email protected]

Translated by ALEX MARTSINKOVSKY































American Mathematical Society · Trudy Moskov. Matem. Obw. Trans. Moscow Math. Soc. Tom 69 (2008)...

Documents

Transcript of American Mathematical Society · Trudy Moskov. Matem. Obw. Trans. Moscow Math. Soc. Tom 69 (2008)...