Toeplitz algebras of Baumslag-Solitar semigroups Astrid an Huef - PowerPoint PPT Presentation

Toeplitz algebras of Baumslag-Solitar semigroups Astrid an Huef Department of Mathematics and Statistics University of Otago, New Zealand Abel Symposium This is a report on joint work with Lisa Orloff Clark and Iain Raeburn, which is to appear in the Indiana Journal of Mathematics.

Let P be a generating subsemigroup of a group G st P ∩ P − 1 = { e } . For x , y ∈ G define x ≤ y ⇐ ⇒ x − 1 y ∈ P . Then ≤ is a partial order on G . Defn. (Nica, 1992) ( G , P ) is quasi-lattice ordered if every pair x , y ∈ G with a common upper bound in P has a least upper bound x ∨ y in P . Equivalently, every element x ∈ G with an upper bound in P has a least upper bound in P (Crisp-Laca, 2002). Write x ∨ y < ∞ if x , y have a least upper bound in P , and x ∨ y = ∞ else. Examples: 1. In ( Z 2 , N 2 ) , have ( m 1 , m 2 ) ≤ ( n 1 , n 2 ) iff m 1 ≤ n 1 and m 2 ≤ n 2 . So ( m 1 , m 2 ) ∨ ( n 1 , n 2 ) = ( max ( m 1 , n 1 ) , max ( m 2 , n 2 )) . 2. ( Q ∗ + , N × ) under multiplication. Here m ≤ n means m | n , and m ∨ n is the lowest common multiple. 3. Let G be the free group with generators a , b , and P the subsemigroup consisting of words in a and b . Then x ≤ y means that x is an initial segment of y , and the rest of y has no factors of a − 1 or b − 1 . Here x ∨ y = ∞ often.

Spse that ( G , P ) is quasi-lattice ordered. Consider ℓ 2 ( P ) with o.n. basis { e x : x ∈ P } . For each x ∈ P , there is an isometry T x on ℓ 2 ( P ) such that T x e y = e xy for y ∈ P . We have T e = 1 and T x T y = T xy . Nica observed � T x − 1 ( x ∨ y ) T ∗ if x ∨ y < ∞ y − 1 ( x ∨ y ) T ∗ x T y = (1) 0 if x ∨ y = ∞ . T is called the Toeplitz representation . An isometric representation satisfying (1) is Nica covariant . ( G , P ) has two C ∗ -algebras: the Toeplitz algebra T ( P ) := � T x : x ∈ P � ⊂ B ( ℓ 2 ( P )) and the universal C ∗ -algebra C ∗ ( G , P ) = span { i ( x ) i ( y ) ∗ : x , y ∈ P } . generated by a universal Nica-covariant representation i : P → C ∗ ( G , P ) . The Toeplitz representation T : P → T ( P ) induces a surjection π T : C ∗ ( G , P ) → T ( P ) ; ( G , P ) is called amenable if π T is faithful.

Let c , d ∈ N + . From now on ( G , P ) denotes the Baumslag-Solitar group G := � a , b : ab c = b d a � , with P the submonoid of G generated by a and b . Theorem (Spielberg, 2012): ( G , P ) is quasi-lattice ordered. Crucial is that each x ∈ G has a unique normal form . For this talk, just need to know the form for elements of P . Write θ : G → Z for the homomorphism such that θ ( a ) = 1 and θ ( b ) = 0. If x ∈ P , then x = b s 0 ab s 1 · · · b s k − 1 ab s k where k = θ ( x ) and 0 ≤ s i < d for all i < k and s k ≥ 0. ◮ If 0 ≤ n < d , then b n + d a = b n ab c .

Recall G := � a , b : ab c = b d a � and θ : G → Z satisfies θ ( a ) = 1 and θ ( b ) = 0. Lemma: Let x , y ∈ P such that x ∨ y < ∞ . 1. If θ ( y ) > θ ( x ) then there exists t ∈ N such that x ∨ y = yb t . 2. If θ ( x ) = θ ( y ) then there exists t ∈ N such that either x ∨ y = x = yb t x ∨ y = y = xb t . or Examples: ◮ Let x = b and y = a . Then x ∨ y = ab c = yb c . ◮ Let x = b and y = ab c + n a . Then x ≤ b d a = ab c ≤ ab c + n a = y . So x ∨ y = y = yb 0 . ◮ If 1 ≤ j < d , then a ∨ b j a = ∞ .

Theorem (CaHR): (G, P) is amenable. Thus C ∗ ( G , P ) = span { T x T ∗ y : x , y ∈ P } .The map θ gives a gauge action γ : T → Aut C ∗ ( G , P ) such that γ z ( T x ) = z θ ( x ) T x . Define α : R → Aut C ∗ ( G , P ) by α t = γ e it . What are the KMS states of the dynamical system ( C ∗ ( G , P ) , α ) ? y ) = e it ( θ ( x ) − θ ( y )) T x T ∗ We have α t ( T x T ∗ y . Thus each T x T ∗ y is y ) = e iz ( θ ( x ) − θ ( y )) T x T ∗ analytic, with α z ( T x T ∗ y . A state ψ of C ∗ ( G , P ) is a KMS β state of ( C ∗ ( G , P ) , α ) for β � = 0 if and only if ψ (( T x T ∗ y )( T p T ∗ q )) = ψ (( T p T ∗ q ) α i β ( T x T ∗ y )) for all x , y , p , q ∈ P .

Suppose ψ is a KMS β state on ( C ∗ ( G , P ) , α ) and fix x , y ∈ P . Using the KMS condition twice gives y ) = e − βθ ( x ) ψ ( T ∗ y T x ) = e − β ( θ ( x ) − θ ( y )) ψ ( T x T ∗ ψ ( T x T ∗ y ) . Hence ψ ( T x T ∗ y ) = 0 unless x ∨ y < ∞ (by Nica covariance) and θ ( x ) = θ ( y ) . If so, y ) = e − βθ ( x ) ψ ( T y − 1 ( x ∨ y ) T ∗ ψ ( T x T ∗ x − 1 ( x ∨ y ) ) . But then either x ∨ y = x or x ∨ y = y , so we have proved half of Propn: A state ψ of ( C ∗ ( G , P ) , α ) is a KMS β state if and only if for all x , y ∈ P we have  e − βθ ( x ) ψ ( T y − 1 x ) if θ ( x ) = θ ( y ) and x ∨ y = x   ψ ( T x T ∗ e − βθ ( x ) ψ ( T ∗ y ) = x − 1 y ) if θ ( x ) = θ ( y ) and x ∨ y = y   0 otherwise. Cor 1: A KMS β state of ( C ∗ ( G , P ) , α ) is determined by its values on T b t for t ∈ N .

Cor 2: 1. Every KMS β state of ( C ∗ ( G , P ) , α ) factors through the quotient by the ideal generated by 1 − T b T ∗ b . 2. If β < ln d , then ( C ∗ ( G , P ) , α ) has no KMS β states. d − 1 d − 1 � � e − β ψ ( 1 ) = e − β d ) ψ ( T b j a T ∗ ( 1 = ψ ( 1 ) ≥ b j a ) = j = 0 j = 0 3. Let I be the ideal generated by d − 1 � 1 − T b T ∗ T b j a T ∗ 1 − b j a . and b j = 0 Then a KMS β state factors through the quotient O ( G , P ) := C ∗ ( G , P ) / I if and only if β = ln d . O ( G , P ) is called the Cuntz algebra , and has been studied by Spielberg (2012) and Katsura (2008).

Theorem (CaHR): Let β > ln d . Spse d does not divide c . There is an explicit affine continuous isomorphism µ �→ ψ µ of the simplex P ( T ) of probability measures on T onto the KMS β simplex of ( C ∗ ( G , P ) , α ) . Idea: To build KMS states we refine a technique of Laca-Raeburn (2010). We exploit that all KMS states think T b is unitary. Let K be the subgroup of G generated by b . Then C ∗ ( K ) ∼ = C ( T ) and states on C ∗ ( K ) are probability measures on T . Given a probability measure µ let W be the repn of K on L 2 ( T , µ ) given by ( W b t f )( z ) = z t f ( z ) , We induce it to a large unitary repn Ind W of G . Restricting ( Ind W ) | P to a suitable invariant (but not reducing) subspace, gives a Nica covariant repn. The KMS state is built using the corresponding repn of C ∗ ( G , P ) .

Recall that every KMS ln d state of C ∗ ( G , P ) factors through the Cuntz algebra O ( G , P ) . Write ¯ T x for the image of T x ∈ C ∗ ( G , P ) in O ( G , P ) . Prop: There is a KMS ln d state ψ on ( O ( G , P ) , α ) such that ψ (¯ T x ¯ y ) = δ x , y e − βθ ( x ) . T ∗ If d does not divide c , then this is the only KMS state on ( O ( G , P ) , α ) .

A. Nica, C ∗ -algebras generated by isometries and Wiener-Hopf operators, J. Operator Theory (1992). J. Crisp and M. Laca, Boundary quotients and ideals of Toeplitz C ∗ -algebras of Artin groups, J. Funct. Anal. (2007). T. Katsura, A class of C ∗ -algebras generalising both graph algebras and homeomorphism C ∗ -algebras IV, pure infiniteness. J. Funct. Anal. (2008). M. Laca and I. Raeburn, Phase transition on the Toeplitz algebra of the affine semigroup over the natural numbers, Adv. Math. (2010). J. Spielberg, C ∗ -algebras for categories of paths associated to the Baumslag-Solitar groups, J. London Math. Soc. (2012).