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−1y ∈ 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 (Z2, N2), have (m1, m2) ≤ (n1, n2) iff m1 ≤ n1 and

m2 ≤ n2. So (m1, m2) ∨ (n1, n2) = (max(m1, n1), max(m2, n2)).

• 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 ℓ2(P) with

• .n. basis {ex : x ∈ P}. For each x ∈ P, there is an isometry Tx
• n ℓ2(P) such that Txey = exy for y ∈ P. We have Te = 1 and

TxTy = Txy. Nica observed T ∗

x Ty =

• Tx−1(x∨y)T ∗

y−1(x∨y)

if x ∨ y < ∞ if x ∨ y = ∞. (1) 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) := Tx : x ∈ P ⊂ B(ℓ2(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 πT : C∗(G, P) → T (P); (G, P) is called amenable if πT is faithful.

Let c, d ∈ N+. From now on (G, P) denotes the Baumslag-Solitar group G := a, b : abc = bda, 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 = bs0abs1 · · · bsk−1absk where k = θ(x) and 0 ≤ si < d for all i < k and sk ≥ 0.

◮ If 0 ≤ n < d, then bn+da = bnabc.

Recall G := a, b : abc = bda 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 = ybt.
• 2. If θ(x) = θ(y) then there exists t ∈ N such that either

x ∨ y = x = ybt

• r

x ∨ y = y = xbt. Examples:

◮ Let x = b and y = a. Then x ∨ y = abc = ybc. ◮ Let x = b and y = abc+na. Then

x ≤ bda = abc ≤ abc+na = y. So x ∨ y = y = yb0.

◮ If 1 ≤ j < d, then a ∨ bja = ∞.

Theorem (CaHR): (G, P) is amenable. Thus C∗(G, P) = span{TxT ∗

y : x, y ∈ P}.The map θ gives a

gauge action γ : T → Aut C∗(G, P) such that γz(Tx) = zθ(x)Tx. Define α : R → Aut C∗(G, P) by αt = γeit. What are the KMS states of the dynamical system (C∗(G, P), α)? We have αt(TxT ∗

y ) = eit(θ(x)−θ(y))TxT ∗ y . Thus each TxT ∗ y is

analytic, with αz(TxT ∗

y ) = eiz(θ(x)−θ(y))TxT ∗ y . A state ψ of

C∗(G, P) is a KMSβ state of (C∗(G, P), α) for β = 0 if and only if ψ((TxT ∗

y )(TpT ∗ q )) = ψ((TpT ∗ q )αiβ(TxT ∗ y ))

for all x, y, p, q ∈ P.

Suppose ψ is a KMSβ state on (C∗(G, P), α) and fix x, y ∈ P. Using the KMS condition twice gives ψ(TxT ∗

y ) = e−βθ(x)ψ(T ∗ y Tx) = e−β(θ(x)−θ(y))ψ(TxT ∗ y ).

Hence ψ(TxT ∗

y ) = 0 unless x ∨ y < ∞ (by Nica covariance) and

θ(x) = θ(y). If so, ψ(TxT ∗

y ) = e−βθ(x)ψ(Ty−1(x∨y)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 KMSβ state if and only if for all x, y ∈ P we have ψ(TxT ∗

y ) =

     e−βθ(x)ψ(Ty−1x) if θ(x) = θ(y) and x ∨ y = x e−βθ(x)ψ(T ∗

x−1y)

if θ(x) = θ(y) and x ∨ y = y

• therwise.

Cor 1: A KMSβ state of (C∗(G, P), α) is determined by its values on Tbt for t ∈ N.

Cor 2:

• 1. Every KMSβ state of (C∗(G, P), α) factors through the

quotient by the ideal generated by 1 − TbT ∗

b .

• 2. If β < ln d, then (C∗(G, P), α) has no KMSβ states.

(1 = ψ(1) ≥

d−1

• j=0

ψ(TbjaT ∗

bja) = d−1

• j=0

e−βψ(1) = e−βd)

• 3. Let I be the ideal generated by

1 − TbT ∗

b

and 1 −

d−1

• j=0

TbjaT ∗

bja.

Then a KMSβ 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 → ψµ of the simplex P(T) of probability measures on T onto the KMSβ 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 Tb 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

• n T. Given a probability measure µ let W be the repn of K on

L2(T, µ) given by (Wbtf)(z) = ztf(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 KMSln d state of C∗(G, P) factors through the Cuntz algebra O(G, P). Write ¯ Tx for the image of Tx ∈ C∗(G, P) in O(G, P). Prop: There is a KMSln d state ψ on (O(G, P), α) such that ψ(¯ Tx ¯ T ∗

y ) = δx,ye−βθ(x).

If d does not divide c, then this is the only KMS state on (O(G, P), α).

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Wiener-Hopf operators, J. Operator Theory (1992).

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algebras and homeomorphism C∗-algebras IV, pure

• infiniteness. J. Funct. Anal. (2008).
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algebra of the affine semigroup over the natural numbers,

• Adv. Math. (2010).
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associated to the Baumslag-Solitar groups, J. London

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