Diffuse traces and Haar unitaries Hannes Thiel TU Dresden Saarbr - - PowerPoint PPT Presentation

Diffuse traces and Haar unitaries

Hannes Thiel

TU Dresden

Saarbr¨ ucken, 11. November 2020

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C*-algebras

Definition A C*-algebra is a norm-closed *-invariant subalgebra A ⊆ B(H) for some Hilbert space H. Examples compact Hausdorff space X C(X) = {f : X → C : f continuous} discrete group G left-regular representation G ℓ2(G) reduced group C*-algebra C∗

red(G) = span{ug : g ∈ G} ⊆ B(ℓ2(G))

C∗

red(Z) ∼

= C(T)

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Traces and tracial states

Definition Let A be a C*-algebra. A trace of A is a positive, linear functional τ : A → C that is tracial: τ(ab) = τ(ba) for all a, b ∈ A. A tracial state is a trace τ with τ = 1. Examples Riesz theorem: Traces on C(X) correspond to positive Borel measures on X. Measure µ corresponds to τµ : C(X) → C, τµ(f) =

• X

f(x)dµ(x), for f ∈ C(X). Tracial states on C(X) probability measures on X Canonical tracial state τG : C∗

red(G) → C,

τG

g∈G

cgug

• = c1,

for cg ∈ C.

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Haar unitaries

Definition Let τ : A → C be a tracial state on a unital C*-algebra A. A unitary u ∈ A is a Haar unitary for τ if τ(uk) = 0 for all k ∈ Z \ {0}. Exercise A unitary u ∈ A is a Haar unitary for τ if and only if sp(u) = T (that is, C∗(u) = C(T)) and the trace τ|C(T) corresponds to the normalized Lebesgue measure λ on T. Example τG : C∗

red(G) → C satisfies τG(ug) =

• 1,

g = 1 0, g = 1 For g ∈ G and k ∈ Z have uk

g = ugk and so τG(uk g) = 0 iff gk = 1.

Thus: ug ∈ C∗

red(G) is a Haar unitary for τG if and only if g has

infinite order.

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Applications of Haar unitaries (1)

Recall that unital subalgebras B, C ⊆ A are free with respect to τ if τ(a1a2 · · · an) = 0 whenever τ(aj) = for all j and either a1, a3, . . . ∈ B and a2, a4, . . . ∈ C or vice versa. Proposition Let p, q ∈ A projections with τ(p) < τ(q). Then p q, if there is Haar unitary u ∈ A such that C∗(p, q, 1) and C∗(u) are free. Proof. If u ∈ A is a Haar unitary, and B ⊆ A unital subalgebra such that C∗(u) and B are free, then B and uBu∗ are free. If ¯ p,¯ q ∈ A are projections such that C∗(¯ p, 1) and C∗(¯ q, 1) are free, and τ(¯ p) < τ(¯ q), then ¯ p ¯ q. Consider ¯ p := p and ¯ q := uqu∗.

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Applications of Haar unitaries (2)

Proposition Let p, q ∈ A projections with τ(p) < τ(q). Then p q, if there is Haar unitary u ∈ A such that C∗(p, q, 1) and C∗(u) are free. Theorem (Dykema-Rørdam 2000) Let (Ak, τk) such that each τk admits a Haar unitary for k ∈ N. If projections p, q in the reduced free product of all (Ak, τk) satisfy τ(p) < τ(q), then p q. Robert 2012: Works also for comparison of positive

• elements. Can compute Cuntz semigroup of C∗

red(F∞), but

not (yet) of C∗

red(F2).

Popa 1995: For every II1 factor M, there exists a Haar unitary u in the ultrapower Mω such that C∗(u) and M are free in Mω.

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The question

Question When does a tracial state admit a Haar unitary?

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The commutative case

Proposition (Dykema-Haagerup-Rørdam 1997) τ : C(X) → C admits a Haar unitary if and only if the associated measure µ on X is diffuse: µ({x}) = 0 for every x ∈ X.

• Proof. (different from DHR)

⇒: Let u ∈ C(X) be a Haar unitary. The inclusion C(T) = C∗(u) ⊆ C(X) corresponds to a surjective map h: X → T such that h∗(µ) = λ. Let x ∈ X. Then µ({x}) ≤ µ(h−1({h(x)})) = λ({h(x)}) = 0. ⇐: Assume µ is diffuse. Sierpi´ nski’s theorem gives Borel sets (Et)t∈[0,1] such that: (a) µ(Et) = t; and (b) Et′ ⊆ Et if t′ ≤ t. Using regularity of the measure can find open sets (Ut)t∈[0,1] such that: (a) µ(Ut) = t; and (b) Ut′ ⊆ Ut if t′ < t; and (c) Ut = {Ut′ : t′ < t}. Get f : X → [0, 1] with Ut = f −1([0, t)). Use u := exp(2πif).

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The main result

Theorem (T-2020) Let (A, τ) be a unital C∗-algebra with a tracial state. TFAE:

1 τ admits a Haar unitary; 2 there exists a (maximal) unital, abelian sub-C∗-algebra

C(X) ⊆ A such that τ induces a diffuse measure on X;

3 τ is diffuse: the unique extension to a normal, tracial state

A∗∗ → C vanishes on every minimal projection in A∗∗;

4 πτ(A)′′ is a diffuse von Neumann algebra; 5 τ does not dominate a trace that factors through a

finite-dimensional quotient of A. (1)⇒(2): consider C(T) ∼ = C∗(u) ⊆ A (3)⇔(4)⇔(5) not so difficult.

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Sketch of the proof

1 τ admits a Haar unitary; 2 there exists a unital, abelian sub-C∗-algebra C(X) ⊆ A

such that τ induces a diffuse measure µ on X;

3 τ is diffuse; 5 τ does not dominate a trace that factors through a

finite-dimensional quotient of A. (2)⇒(5): Let π: A → Mn(C) and c > 0 such that τ ≥ c · trn ◦π. There are x1, . . . , xn ∈ X such that π|C(X) : C(X) → Mn(C) is unitarily conjugate to f → diag(f(x1), . . . , f(xn)). Then µ({x1, . . . , xn}) ≥ c > 0.

• (3)⇒(1): An open projection is p ∈ A∗∗ that is the weak*-limit
• f an increasing net in A+.

Using that τ is diffuse, we construct open projections (pt)t∈[0,1] such that: (a) τ(pt) = t; and (b) pt′ ≤ pt if t′ < t; and (c) pt = sup{pt′ : t′ < t}. contractive a ∈ A+ with pt = ✶[0,t)(a). Use u := exp(2πia).

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Direct consequences

Theorem (T-2020) Let (A, τ) be a unital C∗-algebra with a tracial state. TFAE:

1 τ admits a Haar unitary; 5 τ does not dominate a trace that factors through a

finite-dimensional quotient of A. Corollary A unital C∗-algebra has no finite-dimensional representations if and only if every of its tracial states admits a Haar unitary. Corollary Every tracial state on an infinite-dimensional, simple, unital C∗-algebra admits a Haar unitary.

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Comparison with von Neumann algebras

Proposition Let M be a diffuse von Neumann algebra, and τ : M → C a normal trace. Then every masa of M contains a Haar unitary. The analog is not true for (diffuse) traces on C∗-algebras: Example Let T ⊆ B(ℓ2(N)) Toeplitz algebra. Masa ℓ∞(N) ⊆ B(ℓ2(N)) leads to masa B := ℓ∞(N) ∩ T in T .

K T

π C(T)

c0(N)

• B
• C
• Diffuse trace τ0 on C(T) induces diffuse trace τ := τ0 ◦ π on T .

But B contains no Haar unitary for τ: Given u ∈ B, have π(u) = z ∈ T ⊆ C and then τ(u) = z = 0.

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Application: Group C*-algebras (1)

Let G be a discrete group. Proposition G is infinite if and only if τG : C∗

red(G) → C admits a Haar unitary.

Proof. G infinite ⇔ W∗(G) = πτG(G)′′ diffuse (Dykema 1993) ⇔ τG diffuse ⇔ τG admits Haar unitary Example Let G be an infinite torsion group (e.g. Q/Z). Then τG admits a Haar unitary, but none of the unitaries ug for g ∈ G is Haar.

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Application: Group C*-algebras (2)

Proposition G is infinite if and only if τG admits a Haar unitary. Proposition G is nonamenable if and only if every tracial state of C∗

red(G)

admits a Haar unitary. Proof. G nonamenable ⇔ C∗

red(G) has no finite-dimensional representations

⇔ every tracial state on C∗

red(G) admits Haar unitary

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Application: Free products (1)

We consider unital C∗-algebras with faithful tracial states. The reduced free product of (A, τA) and (B, τB) is the (unique) (C, τC) together with unital embeddings A, B ⊆ C such that

1 τC restricts to τA on A, and to τB on B; 2 C = C∗(A, B); 3 A and B are free with respect to τC.

We write (C, τC) = (A, τA) ∗red (B, τB). Motivating example: (C∗

red(G), τG) ∗red (C∗ red(H), τH) ∼

= (C∗

red(G ∗ H), τG∗H).

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Application: Free products (2)

Example (C∗

red(F2), τF2) ∼

= (C∗

red(Z ∗ Z), τZ∗Z) ∼

= (C(T), µλ) ∗red (C(T), µλ). C∗

red(F2) is simple (Powers 1975).

C∗

red(F2) has stable rank one (Dykema-Haagerup-Rørdam ’97).

Definition (Rieffel 1983) A unital C∗-algebra has stable rank one (SR1) if its invertible elements are dense. Important in K-theory. Recall K0(A) = Gr(V(A)) for V(A) =

• finitely generated, projective A-modules
• / ∼

= . If A has SR1, then V(A) is cancellative and then V(A) ⊆ K0(A). Have: K0(C∗

red(F2)) ∼

= Z. It follows that V(C∗

red(F2)) ∼

= N. Every finitely generated, projective C∗

red(F2)-module is free.

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Application: Free products (3)

Theorem Let (A, τA) and (B, τB) be unital, simple C∗-algebras with stable rank one. Then so is (A, τA) ∗red (B, τB). Proof. If A = C or B = C, then reduced free product is B or A. So assume A, B = C. Case 1: A and B finite-dimensional (Avitzour 1982, DHR 1997) Case 2: A or B infinite-dimsional. Then τA or τB admits Haar

• unitary. Can apply result of Dykema 1999.

This is the C∗-analog of: reduced free product of II1-factors is again II1-factor. (We consider infinite-dimensional, unital, simple C∗-algebras with stable rank one as C∗-analogs of II1-factors.)

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Questions

Question 1 When is there u ∈ A with τ(u) = τ(u2) = . . . = τ(uk) = 0? Does every τ admit such u if and only if A has no k-dimensional representations? Already interesting for k = 1. Question 2 If ϕ: A → C is a state, the centralizer is Aϕ := {a ∈ A : ϕ(ab) = ϕ(ba) for all b ∈ A}. When does Aϕ contain a Haar unitary?

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Thank you.

Reference:

• Thiel. Diffuse traces and Haar unitaries. arXiv:2009.06940,

25 pp.

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