Uniqueness Theorem

Uniqueness theorems for combinatorialC-algebras

Sarah Reznikoff

joint work with Jonathan H. Brown, Gabriel Nagy, Aidan Sims, and Dana Williamsfunded in part by NSF DMS-1201564

Nebraska-Iowa Functional Analysis Seminar 2015Creighton University

Introductionk -graph algebras

History of uniqueness theoremsAperiodicity

Classical theoremsRecent workWhy graph algebras?

Let G be a graph, k -graph, or groupoid, and C(G ) theuniversal C*-algebra defined from it.

Question: Under what circumstances is a -homomorphism : C(G ) B(H) injective?Classical theorems addressing this question assume either(a) the existence of intertwining gauge actions" on the

algebras (Gauge Invariant Uniqueness Theorem 1), or(b) an aperiodicity condition on the graph itself (Cuntz-Krieger

Uniqueness Theorem 2),and conclude that is injective iff it is nondegenerate, i.e.,injective on the diagonal subalgebra D.

1an-Huef, 972Fowler-Kumjian-Pask-Raeburn, 97

Sarah Reznikoff Uniqueness theorems for combinatorial C-algebras




Theorem (Brown-Nagy-R-Sims-Williams) There is a canonicalsubalgebraM C(G ) such that a -homomorphism : C(G ) B(H) is injective iff |M is injective.Moreover,M C(G ) is a Cartan inclusion.

[NR1] Nagy and Reznikoff, Abelian core of graph algebras,J. Lond. Math. Soc. (2) 85 (2012), no. 3, 889908.[NR2] Nagy and Reznikoff, Pseudo-diagonals and uniquenesstheorems, Proc. AMS (2013).[BNR] Brown, Nagy, Reznikoff A generalized Cuntz-Kriegeruniqueness theorem for higher-rank graphs, JFA (2013).[BNRSW] Brown, Nagy, Reznikoff, Sims, and Williams, Cartansubalgebras of groupoid C-algebras (2015).





Drinen (1999): Every AF algebra is Morita equivalent to agraph algebra.

Kumjian-Pask-Raeburn (1998)I The algebra is AF iff the graph has no cycles.I All simple graph algebras are AF or purely infinite.

Hong-Szymanski (2004): the ideal structure of the algebra canbe completely described from the graph.

Generalizations and related constructions: Exel crossedproduct algebras, Leavitt path algebras (Abrams, Ruiz,Tomforde), topological graph algebras (Katsura), Ruellealgebras (Putnam, Spielberg), Exel-Laca algebras, ultragraphs(Tomforde), Cuntz-Pimsner algebras, higher-rankCuntz-Krieger algebras (Robertson-Steger), etc.





k -graph algebras (Kumjian and Pask, 2000)

developed to generalize graph algebras and higher-rankCuntz-Krieger algebras, whether simple, purely infinite, or AF can be determined fromproperties of the graph (Kumjian-Pask, Evans-Sims), are groupoid C*-algebras, include examples of algebras that are simple but neither AFnor purely infinite, and hence not graph algebras(Pask-Raeburn-Rordam-Sims), include examples that can be constructed from shift spaces(Pask-Raeburn-Weaver), can be used to construct any Kirchberg algebra (Spielberg).




DefinitionExample: kCuntz-Krieger families

Let k N+. We regard Nk as a category with a single object, 0,and with composition of morphisms given by addition.

A k -graph is a countable category along with a degreefunctor d : Nk satisfying the unique factorization property:

For all , and m, n Nk , if d() = m + n then there areunique d1(m) and d1(n) such that = .

I Denote the range and source maps r , s : .I Refer to objects as vertices and morphisms as paths.I We assume: for all v , n Nk ,

0 < |r1({v}) d1({n})|




Example: Rectangles in NkLet k := {(l ,n) Nk Nk | l n} with d(l ,n) = n l ,s(m, l) = l = r(l ,n), and (m, l)(l ,n) = (m,n).

0

m

nl

n

m





A Cuntz-Krieger -family in a C*-algebra A is a set{T, } of partial isometries in A satisfying

(i) {Tv | v d1({0}) is a family of mutually orthogonalprojections,

(ii) T = TT for all , s.t. s() = r(),(iii) T T = Ts() for all , and(iv) for all v d1({0}) and n Nk , Tv =

d()=nr()=v

TT .

C() will denote the C*-algebra generated by a universalCuntz-Krieger -family, (S, ), with P = SS.Note: C() = span{SS |, | s() = s()}Defn: The diagonal D := span{SS | }.




Classic theoremsNon-degeneracy does not suffice

Classic uniqueness theorems

Coburns Theorem (67) ef

Any nondegenerate representation C(Te,Tf ) is isomorphic tothe Toeplitz algebra T , generated by one non-unitary isometry.

Cuntz (77) Any nondegenerate representation isisomorphic to the Cuntz AlgebraOn, generated by n

partial isometries Si satisfying i , Si Si =n

j=1

SjSj .

n loops

Cuntz-Krieger (80) Graph with 0-1 adjacency matrix AWhen A satisfies a fullness condition (I), any nondegeneraterepresentation is isomorphic to the Cuntz-Krieger algebra OA.




Classic theoremsNon-degeneracy does not suffice

Is non-degeneracy enough?

No! Consider the cycle of length three, E .The map : C(E) M3(C) given by

Svi 7 i,i Sei,j 7 j,i .

is a -homomorphism. However

(Se3,1Se2,3Se1,2) = 1,33,22,1 = 1,1 = (Sv1),

whereas Se3,1Se2,3Se1,2 6= Sv1 in C(E), so is not injective.

v1

v2 E

v3e3,1

e2,3

e1,2




GroupoidsThe C*-algebra of a groupoidState extensions

The infinite path space

Defn. An infinite path in a k -graph is a degree-preservingcovariant functor x : k .

k = 1 picture e1 e2x(1,3) = e1e2d(e1e2) = 2

k = 2 picture

r()

s() x((1,2), (3,3)) = d() = (2,1)





An infinite path x in a k -graph is eventually periodic if there are 6= in and y such that x = y = y ; otherwise x isaperiodic.

x

y

y





is aperiodic if every vertex is the range vertex of an aperiodicinfinite path. In a directed graph, cycles without entry reveal failure ofaperiodicity.

v

Clearly the only infinite path with range v , , is eventually periodic.

A k -graph is aperiodic iff D is a masa.Cuntz-Krieger Uniqueness Theorem:(Kumjian-Pask-Raeburn-Fowler, et. al. (90s))When is nondegenerate and the graph satisfies

(L) every cycle has an entrythen is injective.





Theorem Szymanski (2001), Nagy-R (2010):Condition (L) can be replaced with a condition on the spectrumof (S) for cycles without entry.

Uniqueness theorems of Raeburn-Sims-Yeend andKumjian-Pask assume aperiodicity of the k -graph.

Theorem Nagy-Brown-R (2013, JFA)A -homomorphism : C() A is injective iff it is injectiveon the subalgebraM := C(SS | = }.Theorem Yang (2014)I M = D.I For fixed m,n Nk , and 0 with d() = n, and an

element X =

SS ofM , there is at most one termS0S

0

with d(0) = m.





A groupoid G is a small category in which every element hasan inverse. When G is a topological groupoid, C(G) is definedto be a completion of Cc(G).The isotropy subgroupoid is the setIso(G) := {g G | r(g) = s(g)}Theorem (Brown-Nagy-R-Sims-Williams, 2014)Let G be a locally compact, amenable, Hausdorff, talegroupoid, with (IsoG) is closed. If : C(G) A is aC-homomorphism, then the following are equivalent.

(i) is injective.(ii) is injective onM := C((Iso(G))).





Assume G is a 2nd countable locally compact Hausdorff talegroupoid. For f Cc(G), define

f () = f (1) f g () ==

f ()g().

Define the I-norm on Cc(G) by

fI = supuG(0)

max

Gu

|f ()|,Gu

|f ()| , and let

f = sup{pi(f ) |pi is an I-norm bounded -rep. of Cc(G)}C(G) is the completion of Cc(G) with respect to this norm.





Given a k -graph , define

G = {(y ,d , y) | y , , , s() = s() = r(y),d = d() d()}

withr(x ,d , y) = x , s(x ,d , y) = y

(x ,d , y)(z,d ,w) = y (z)(x ,d + d ,w).

The cylinder sets Z (, ) = {(y ,d , z) G} form a basisfor an tale groupoid. Moreover, C() = C(G) andM = C((Iso(G))) via themap SS 7 Z (,).





(Renault, 80) A masa C*-subalgebra B A is Cartan if(i) a faithful conditional expectation A B,(ii) The normalizer of B in A generates A, and(iii) B contains an approximate unit of A.Extension properties for pure states on masa B A:(UEP) Every pure state extends uniquely to A.(AEP) Densely many pure states extend uniquely.

Kadison-Singer Problem (Marcus-Spielman-Srivastava)(UEP) holds for all masa in B(`2(N)).

Thm (Nagy-R, 2011; NRBSW, 2014)M C(G) is Cartan.Thm (NRBSW, 2014) All Cartan subalgebras satisfy the AEP.





Thank you!





A. an Huef and I. Raeburn, The ideal structure ofCuntz-Krieger algebras, Ergodic Theory Dynam. Systems17 (1997), 611624.

J.H. Brown, G. Nagy, and S. Reznikoff, A generalizedCuntz-Krieger uniqueness theorem for higher-rank graphs,J. Funct. Anal. (2013),http://dx.doi/org/10.1016/j.jfa.2013.08.020.

J.H. Brown, G. Nagy, S. Reznikoff, A. Sims, andD. Williams, Cartan subalgebras of groupoid C-algebras

K.R. Davidson, S.C. Power, and D. Yang, Dilation theory forrank 2 graph algebras, J. Operator Theory.





P. Goldstein, On graph C*-algebras, J. Austral. Math. Soc.72 (2002), 153160

A. Kumjian and D. Pask, Higher rank graph C*-algebras,New York J. Math. 6 (2000), 120.

A. Kumjian, D. Pask, and I. Raeburn, Cuntz-Kriegeralgebras of directed graphs, Pacific J. Math. 184 (1998)161174.

A. Kumjian, D. Pask, I. Raeburn, and J. Renault, Graphs,groupoids and Cuntz-Krieger algebras, J. Funct. Anal. 144(1997), 505541

G. Nagy and S. Reznikoff, Abelian core of graph algebras,J. Lond. Math. Soc. (2) 85 (2012), no. 3, 889908.





G. Nagy and S. Reznikoff, Pseudo-diagonals anduniqueness theorems, (2013), to appear in Proc. AMS.

D. Pask, I. Raeburn, M. Rrdam, A. Sims, Rank-twographs whose C*-algebras are direct limits of circlealgebras, J. Functional Anal. 144 (2006), 137178.

I. Raeburn, A. Sims and T. Yeend, Higher-rank graphs andtheir C*-algebras, Proc. Edin. Math. Soc. 46 (2003) 99115.

D. Robertson and A. Sims, Simplicity of C*-algebrasassociated to higher-rank graphs. Bull. Lond. Math. Soc. 39(2007), no. 2, 337344.

G. Robertson and T. Steger, Affine buildings, tiling systemsand higher rank Cuntz-Krieger algebras, J. ReineAngew. Math. 513 (1999), 115144.





A. Sims, Gauge-invariant ideals in the C*-algebras offinitely aligned higher-rank graphs, Canad. J. Math. 58(2006), no. 6, 12681290.

J. Spielberg, Graph-based models for Kirchberg algebras,J. Operator Theory 57 (2007), 347374.

W. Szymanski, General Cuntz-Krieger uniquenesstheorem, Internat. J. Math. 13 (2002) 549555.

D. Yang, Cycline subalgebras are Cartan, preprint.

T. Yeend, Groupoid models for the C*-algebras oftopological higher-rank graphs, J. Operator Theory 57:1(2007), 96120


IntroductionClassical theoremsRecent workWhy graph algebras?

k-graph algebrasDefinitionExample: kCuntz-Krieger families

History of uniqueness theoremsClassic theoremsNon-degeneracy does not suffice

AperiodicityGroupoidsThe C*-algebra of a groupoidState extensions

Uniqueness Theorem

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