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 1 / 19

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 � ℓ 2 ( G ) � reduced group C*-algebra red ( G ) = span { u g : g ∈ G } ⊆ B ( ℓ 2 ( G )) C ∗ red ( Z ) ∼ C ∗ = C ( T ) 2 / 19

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 � τ µ : C ( X ) → C , τ µ ( f ) = f ( x ) d µ ( x ) , for f ∈ C ( X ) . X Tracial states on C ( X ) � probability measures on X Canonical tracial state � � τ G : C ∗ � red ( G ) → C , τ G c g u g = c 1 , for c g ∈ C . g ∈ G 3 / 19

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 ( u k ) = 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 � 1 , g = 1 τ G : C ∗ red ( G ) → C satisfies τ G ( u g ) = 0 , g � = 1 g ) = 0 iff g k � = 1 . For g ∈ G and k ∈ Z have u k g = u g k and so τ G ( u k Thus: u g ∈ C ∗ red ( G ) is a Haar unitary for τ G if and only if g has infinite order. 4 / 19

Applications of Haar unitaries (1) Recall that unital subalgebras B , C ⊆ A are free with respect to τ if τ ( a 1 a 2 · · · a n ) = 0 whenever τ ( a j ) = for all j and either a 1 , a 3 , . . . ∈ B and a 2 , a 4 , . . . ∈ 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 ∗ . 5 / 19

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 ( A k , τ k ) such that each τ k admits a Haar unitary for k ∈ N . If projections p , q in the reduced free product of all ( A k , τ k ) 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 ( F 2 ) . Popa 1995: For every II 1 factor M , there exists a Haar unitary u in the ultrapower M ω such that C ∗ ( u ) and M are free in M ω . 6 / 19

The question Question When does a tracial state admit a Haar unitary? 7 / 19

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 ( E t ) t ∈ [ 0 , 1 ] such that: (a) µ ( E t ) = t ; and (b) E t ′ ⊆ E t if t ′ ≤ t . Using regularity of the measure can find open sets ( U t ) t ∈ [ 0 , 1 ] such that: (a) µ ( U t ) = t ; and (b) U t ′ ⊆ U t if t ′ < t ; and (c) U t = � { U t ′ : t ′ < t } . � Get f : X → [ 0 , 1 ] with U t = f − 1 ([ 0 , t )) . Use u := exp( 2 π if ) . 8 / 19

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 π τ ( 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. 9 / 19

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 → M n ( C ) and c > 0 such that τ ≥ c · tr n ◦ π . There are x 1 , . . . , x n ∈ X such that π | C ( X ) : C ( X ) → M n ( C ) is unitarily conjugate to f �→ diag( f ( x 1 ) , . . . , f ( x n )) . Then µ ( { x 1 , . . . , x n } ) ≥ c > 0 . � (3) ⇒ (1): An open projection is p ∈ A ∗∗ that is the weak*-limit of an increasing net in A + . Using that τ is diffuse, we construct open projections ( p t ) t ∈ [ 0 , 1 ] such that: (a) τ ( p t ) = t ; and (b) p t ′ ≤ p t if t ′ < t ; and (c) p t = sup { p t ′ : t ′ < t } . � contractive a ∈ A + with p t = ✶ [ 0 , t ) ( a ) . Use u := exp( 2 π ia ) . 10 / 19

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. 11 / 19

� � � � � � � � � � � � 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 ( ℓ 2 ( N )) Toeplitz algebra. Masa ℓ ∞ ( N ) ⊆ B ( ℓ 2 ( N )) leads to masa B := ℓ ∞ ( N ) ∩ T in T . π � C ( T ) � K � T � 0 0 � c 0 ( N ) 0 B 0 C Diffuse trace τ 0 on C ( T ) induces diffuse trace τ := τ 0 ◦ 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 . � 12 / 19

Application: Group C*-algebras (1) Let G be a discrete group. Proposition G is infinite if and only if τ G : C ∗ red ( G ) → C admits a Haar unitary. Proof. G infinite ⇔ W ∗ ( G ) = π τ G ( G ) ′′ diffuse (Dykema 1993) ⇔ τ G diffuse ⇔ τ G admits Haar unitary Example Let G be an infinite torsion group (e.g. Q / Z ). Then τ G admits a Haar unitary, but none of the unitaries u g for g ∈ G is Haar. 13 / 19

Application: Group C*-algebras (2) Proposition G is infinite if and only if τ G 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 14 / 19

Application: Free products (1) We consider unital C ∗ -algebras with faithful tracial states. The reduced free product of ( A , τ A ) and ( B , τ B ) is the (unique) ( C , τ C ) together with unital embeddings A , B ⊆ C such that 1 τ C restricts to τ A on A , and to τ B on B ; 2 C = C ∗ ( A , B ) ; 3 A and B are free with respect to τ C . We write ( C , τ C ) = ( A , τ A ) ∗ red ( B , τ B ) . Motivating example: red ( H ) , τ H ) ∼ ( C ∗ red ( G ) , τ G ) ∗ red ( C ∗ = ( C ∗ red ( G ∗ H ) , τ G ∗ H ) . 15 / 19

Application: Free products (2) Example red ( F 2 ) , τ F 2 ) ∼ red ( Z ∗ Z ) , τ Z ∗ Z ) ∼ ( C ∗ = ( C ∗ = ( C ( T ) , µ λ ) ∗ red ( C ( T ) , µ λ ) . C ∗ red ( F 2 ) is simple (Powers 1975). C ∗ red ( F 2 ) 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 K 0 ( 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 ) ⊆ K 0 ( A ) . red ( F 2 )) ∼ red ( F 2 )) ∼ Have: K 0 ( C ∗ = Z . It follows that V ( C ∗ = N . � Every finitely generated, projective C ∗ red ( F 2 ) -module is free. 16 / 19