Noncommutative Potential Theory onC∗-algebras

Fabio Cipriani

Dipartimento di MatematicaPolitecnico di Milano(

Joint works with J.-L. Sauvageot)

S.I.S.S.A. Trieste, May 19, 2011

Our aim is to show how energy or Dirichlet forms (E ,F) determine and influence thePotential Theory of Noncommutative spaces represented by a C∗-algebra with trace(A, τ), with special attention to

Spectral Triples (A,D,H)

Finite Energy states ant their potentials on A

Multipliers of (E ,F)

A special role will be played by the energy functionals Γ[a] of elements a ∈ F .

Classical Potential Theory

A fruitful approach to CPT focuses around the notion of Energy:

X loc. compact metrizable space

m positive Radon measure a Dirichlet or Energy form E : L2(X,m)→ [0,+∞]is a l.s.c. quadratic functional satisfying the contraction property

E [a ∧ 1] ≤ E [a] a ∈ L2(X,m) and such that the

Dirichlet or Energy space F := a ∈ L2(X,m) : E [a] < +∞ endowed withits

graph norm: ‖a‖F :=√‖a‖2

2 + E [a] is

regular: the Dirichlet algebra B : F ∩C0(X) is dense in L2(X,m) and in C0(X).

The capacity associated to (E ,F)

Cap(O) := inf‖a‖2F : a = 1 on O O ⊆ X open

Cap(B) := infCap(O) : B ⊆ O open B ⊆ X

endows X with a finer notion of smallness of subsets called polarity.

C∗-algebras and traces

A C∗-algebra A is an involutive Banach algebra such that

‖a∗a‖ = ‖a‖2 a ∈ A.

A trace τ : A→ C is a linear functional such that

τ(a∗a) ≥ 0 , τ(ab) = τ(ba) .

As a fundamental noncommutative example one may consider the algebra B(h) of allbounded operators on a Hilbert space h and the usual trace defined on it.

Considering the quotient of A with respect to the scalar product (a, b) = τ(a∗b) andcompleting, one gets a Hilbert space L2(A, τ) as well a continuous morphism ofinvolutive algebras (called representation)

πτ : A→ B(L2(A, τ))

from the regular representation of A onto itself: π : τ(a)b := ab.

Examples

C0(X) algebra of continuous function vanishing at infinity on a l.c. space

Matrix algebras Mn(C) and their traces Trn(a) =∑n

i=1 aii

Algebra of bounded operators B(h) in a Hilbert space h

any closed ∗-subalgebra A ⊂ B(h)

the closed ideal of compact operators K(h) in a Hilbert space h

Calkin algebra B(h)/K(h)

Group C∗-algebra C∗(G) of a locally compact group

Crossed product C∗-algebras C0(X) oα G of dynamical systems (α,G,X)

C∗-algebras of foliations

C∗-algebra of orbifolds: A = a ∈ C(X,Mn(C)) : a(x1) , · · · a(xr) diagonalwith underlying topological space X and elliptic points x1, · · · , xr ∈ X.

Dirichlet forms on (A, τ)

The generalization of the notion of energy to the noncommutative setting is given by

A C∗-Dirichlet form E : L2(A, τ)→ [0,+∞], a l.s.c. quadratic functional satisfying

reality: E [a∗] = E [a]

regularity: the Dirichlet algebra B := F ∩ A is a form core dense in A

complete Markovianity:

En[a ∧ 1] ≤ En[a] L2(Mn(A), τ ⊗ tr)), n ≥ 1

where En denotes the ampliation of E from (A, τ) to (Mn(A), τ ⊗ tr).

Markovian semigroups on (A, τ)

A Markovian semigroup Tt : L2(A, τ)→ L2(A, τ) on (A, τ) is a stronglycontinuous, symmetric, positive, contraction semigroup such that

0 ≤ a ≤ 1 ⇒ 0 ≤ Tta ≤ 1 .

A Markovian semigroup extends to a weakly∗-continuously to a positive, contractionsemigroup on the von Neumann algebra L∞(A, τ).

A C∗-Markovian semigroup is required to satisfies these properties with respect toall its ampliations to (A⊗Mn(C), τ ⊗ Trn).

Theorem. (Albeverio-Hoegh.Krhon 1976)

C∗-Dirichlet forms and C∗-Markovian semigroups are in one-to-one correspondenceby their generators, i.e. the positive self-adjoint operators (L,D(L)) such that

E [a] = ‖L1/2a‖22 F = D(L1/2) , Tt = e−tL .

Examples on groups C∗-algebras

Let G be a locally compact, unimodular group,

L1(G) be the convolution algebra with involution

(f ∗ g)(t) =

∫G

ds f (s)g(s−1t) , f ∗(t) = f (t−1) ,

λ : L1(G)→ B(L2(G)) the left regular representation λ(f )g = f ∗ g,

C∗r (G) := λ(L1(G)) the reduced C∗-algebra of G,

τ(a) := a(e) the trace on C∗r (G) initially defined for a ∈ Cc(G)

` : G→ C a continuous function of negative type.

Then

E [a] :=∫

G ds `(s)|a(s)|2 is a C∗-Dirichlet form

Tta := e−t`a is the corresponding C∗-Markovian semigroup.

Energy functional or carre du champ

Theorem. (F-C, J.L. Sauvageot 2003)

Let (E ,F) be a C∗-Dirichlet form on (A, τ) and B = F ∩ A the associated Dirichletalgebra. Then for any fixed a, b ∈ B the functional

〈Γ(a, b), c〉 = c ∈ B

extends to a continuous functional on A. In particular Γ[a] := Γ(a, a) is a positivefunctional, called energy functional or carre du champ, such that

E [a] = 〈Γ[a], 1〉 .

Γ[a] represents the energy distribution of the configuration a ∈ F .

Construction and representation of Dirichlet forms by Derivations

Definition. (F.C.-J.-L. Sauvageot)

A derivation (B, ∂,H,J ) on A is closed linear map ∂ : B → H such that

B ⊂ A is a dense ∗-sub-algebra

(H,J ) ia symmetric Hilbert A-bimodule

∂ is symmetric J (∂(a∗)) = ∂(a∗) and satisfies the Leibniz rule

∂(ab) = (∂a)b + a(∂b) a, b ∈ B .

Theorem. (F.C.-J.-L. Sauvageot)

Let (B, ∂,H,J ) be a derivation on A. The closure of the quadratic form

E [a] := ‖∂a‖2 a ∈ B

is then a C∗-Dirichlet form.

Any C∗-Dirichlet form (E ,F) can be represented uniquely by a suitablederivation on the Dirichlet algebra B = F ∩ A.

The generator associated with (E ,F) can be written as L = ∂∗ ∂.

Example: derivation associated to a function of negative type

Any continuous function of negative type ` : G→ C on locally compact group G,can be represented as

`(t) = ‖c(t)‖2 t ∈ G

for a continuous 1-cocycle c : G→ K

c(st) = c(s) + π(s)c(t) s, t ∈ G

of a suitable unitary representation π : G→ K.

Then the Hilbert spaceH := K ⊗ L2(G) is a C∗r (G)-bimodule with respect to theright id⊗ ρ and left π ⊗ λ actions (ρ and λ being the right and left regularrepresentations of G). The linear map

∂(a)(t) := a(t)c(t) a ∈ L1(G) t ∈ G

is a derivation such that E [a] :=∫

G ds `(s)|a(s)|2 = ‖∂a‖2

Derivations and energy functionals

Theorem. (F.C.-J.-L. Sauvageot)

The following relation

((∂c)d|(∂a)b)H = 〈Γ(c, a), bd∗〉 a, b, c, d ∈ B

holds true between derivation and energy functional associated to a C∗-Dirichletform. It implies that the carre du champ satisfies a Leibniz rule too.

Energy functionsls and Spectral TripleGiven a C∗-Dirichlet form (E ,F) with Dirichlet algebra B on the C∗-algebra withtrace (A, τ), a natural candidate for a Spectral Triple (A,D,K) is given by

A := B , D :=

(0 ∂∂∗ 0

), K := H⊕ L2(A, τ)

Theorem. (F.C.-J.-L. Sauvageot 2011)

The following conditions are equivalent for a ∈ B:

the commutator [D, a] is bounded on Kthe energy functional Γ[a] is a coefficient of the bimoduleHthe Hilbert space L2(A,Γ[a]) is embedded inH as a sub-bimodule.

If D(L) is the domain of the generator associated to the Dirichlet form and

a, a2 ∈ D(L)

the above conditions are verified. As a consequence, if the spectrum of (L,D(L) isdiscrete and D(L) contains a dense sub-algebra A closed by holomorphic functionalcalculus, then (A,D,K) is a Spectral Triple.

Crossed product C∗-algebra of dynamical systems

Let G ⊆ Isom(M) be a unimodular group of isometries of a Riemannian manifold Mand consider the crossed product C∗-algebra A = C0(M) o G

a ∗ b(x, t) =

∫G

a(x, s)b(xs, s−1t)ds product

a∗(x, t) = a(xt, t−1) involution

τ(a) =

∫M

a(x, e)dx trace

L2(A, τ) = L2(M × G) τ(b∗ ∗ a) = (b|a)L2(M×G) GNS− space

Dirichlet form and Spectral Triples on crossed product of dynamical systems

Theorem. (F.C.-Mauri 2009)

Let ` : G→ [0,+∞) be a negative type function. Then the operator∆ : Cc(G,C∞c (M))→ L2(M × G)

∆a(x, t) = (∆Ma(t))(x) + l(t)a(x, t)

admits a self-adjoint, nonnegative extension associated to the Dirichlet formE : L2(M × G)→ [0,+∞]

E [a] =

∫G|∇Ma(t)|2L2(TM) + l(t)‖a(t)‖2

L2(M)

generating a Markovian semigroup e−t∆ on C0(M) o G.

The proof proceeds through the construction of a suitable derivation on thec∗-algebra C0(M) o G.

Finite Energy Functionals and Potentials

Let (E ,F) be C∗-Dirichlet form on (A, τ) and consider norm ‖a‖2F := ‖a‖2

2 + E [a].

Definition. (C-Sauvageot 2010)

ω ∈ A∗+ will be said a finite energy functional if there exists Cω > 0 such that

|ω(b)| ≤ Cω‖b‖F b ∈ B .

An element g ∈ F is said to be a potential if

a ∈ B+ ⇒ (a|g)F ≥ 0 .

Example. Let h ∈ L2+(A, τ) ∩ L1(A, τ). Then the positive linear form

ωh(b) := τ(hb) b ∈ A

is a finite energy functional and its potential is given by

g(ωh) = (I + L)−1h ∈ F+ .

Maximum Principle

Thm. (C-Sauvageot 2010)

1. Finite Energy Functionals and Potentials are in one-to-one correspondence by

ω(b) = (g(ω)|b)F b ∈ B+ .

The energy of ω is defined to be E [ω] := ‖g(ω)‖2F .

2. Let ω, ω′ ∈ A∗∗ be such that ω′ ≤ ω and ω has finite energy. Then ω′ has finiteenergy and

g(ω′) ≤ g(ω) , E [ω′] ≤ E [ω] .

In particular, potentials are positive: g ∈ F+.

Example: Potential Theory on group C∗-algebras

Let G be a loc. compact group with the Haagerup property and l : G→ [0,+∞) aproper continuous function of negative type. Then

E [a] :=

∫G

l(g)|a(g)|2 dg

is a C∗-Dirichlet form on the group C∗-algebra C∗r (G) ([C-Sauvageot 2001]).

1. When G is discrete, finite energy functionals are characterized by the squareintegrability of the function

g 7→ ξ(g) :=ω(δg)√1 + l(g)

which is then a coefficient of the left regular representation.

Example: Potential Theory on group C∗-algebras

2. Let ξ ∈ L2(G) and ωξ ∈ C∗r (G)+ the associated coefficient of the left regularrepresentation

ωξ(a) := (ξ|λ(a)ξ) a ∈ C∗r (G) .

If ωξ has finite energy then its potential is given by

g(ωξ) =ξ∗ ∗ ξ1 + l

.

3. If the length function l : G→ [0,+∞) of a finitely generated discrete group G isof negative type, then the trivial representation π0 never has finite energy.

4. If G = Zn and l(g) := |g|2, then the trivial representation π0 has finite energy ifand only if n = 1.

Noncommutative Deny Theorem

Theorem. (C.Sauvageot 2010)

Let ω ∈ A∗+ be a finite energy functional such that its associated potential is bounded

g(ω) ∈M.

Then the following bound holds true

ω(b∗b) ≤ 2‖g(ω)‖M‖b‖2F b ∈ B .

Hence, there exists a continuous embedding T : F → L2(A, ω) with norm less orequal to

√2‖g(ω)‖M, such that Tb = bΩ where Ω ∈ L2(A, ω) is the canonical

cyclic vector representing ω. If ω is faithful one has

‖∆1/4ω bΩ‖2 ≤ 2‖g(ω)‖M‖b‖2

F b ∈ B

where ∆ω is the Tomita modular operator.

Making use of the Leibniz property of the derivation (B, ∂,H,J ) associated to theDirichlet form we have

Theorem. (C.Sauvageot 2010)

The energy functional Γ(g) ∈ A∗+ of a bounded potential g ∈M is a finite energyfunctional

|〈Γ(g); b〉| ≤ 2‖g‖M‖g‖F‖b‖F b ∈ F .

Noncommutative Mokobodzki Theorem

Definition. (C.Sauvageot 2010)

An element of the von Neumann algebra p ∈M is a multiplier of the Dirichlet space(E ,F) whenever

b ∈ F ⇒ pb, bp ∈ F .

Theorem. (C.Sauvageot 2010)

Let g ∈M+ be a bounded potential such that its energy functional Γ(g) has abounded potential. Then g is multiplier of the Dirichlet space with

‖pb‖F ≤(‖p‖M + ‖g(Γ(p))‖1/2

M

)‖b‖F b ∈ F .

In particular, if h ∈M∩ L2(A, τ) then g := (I + L)−1h is a multiplier.