The $K$-theory of Toeplitz $C^*$-algebras of right-angled Artin groups

Let P be a semigroup that admits an embedding into a group G. Assume that the embedding satisfies the Toeplitz condition of [24] and that the Baum-Connes conjecture holds for G. We prove a formula describing the Ktheory of the reduced crossed product A ⋊α,r P by any automorphic action of P . This formula is obtained as a consequence of a result on the K-theory of crossed products for special actions of G on totally disconnected spaces. We apply our result to various examples including left Ore semigroups and quasilattice ordered semigroups. We also use the results to show that for certain semigroups P , including the ax + b-semigroup R ⋊ R × for a Dedekind domain R, the K-theory of the left and right regular semigroup C*-algebras C * λ (P ) and C * ρ (P ) coincide, although the structure of these algebras can be very different. Preliminaries on totally disconnected spacesRecall that a locally compact Hausdorff space Ω is totally disconnected if and only if its topology has a basis of compact open subsets. The corresponding algebras C 0 (Ω) of continuous functions which vanish at infinity are precisely the commutative AF-Algebras. In what follows, if V ⊆ Ω, then 1 V : Ω → C denotes the characteristic function of V . Definition 2.1. Let Ω be a totally disconnected locally compact Hausdorff space and let V be a family of compact open subsets in Ω. Moreover, let U c (Ω) denote the set of all compact open subsets of Ω. Then we say that V is a generating family of the compact open sets of Ω if U c (Ω) coincides with the smallest family U of compact open sets in Ω which contains V and which is closed under finite intersections, finite unions, and under taking differences U W with U, W ∈ U . Lemma 2.2. Suppose that V is a family of compact open sets in the totally disconnected space Ω. Then the following are equivalent (1) The set {1 V : V ∈ V} generates C 0 (Ω) as a C*-algebra. (2) The set V generates U c (Ω) in the sense of Definition 2.1. Moreover, if V is closed under taking finite intersections, then (1) and (2) are equivalent to(3) span{1 V : V ∈ V} is a dense subalgebra of C 0 (Ω) containing span{1 U : U ∈ U c (Ω)}.

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“…Acknowledgements: We are grateful to Marcelo Laca for drawing our attention to [18] and to Mikael Rørdam for pointing out Example 3.20.…”

Section: Introductionmentioning

confidence: 99%

On the K-Theory of Crossed Products by Automorphic Semigroup Actions

Cuntz¹,

Echterhoff²,

Li³

2013

The Quarterly Journal of Mathematics

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“…The groups A Γ are called right-angled Artin groups and the monoids A + Γ are called right-angled Artin monoids. Their C*-algebras are discussed in [CL02,CL07,Iva10,ELR16].…”

Section: Examplesmentioning

confidence: 99%

Semigroup C*-algebras

Li¹

2017

Preprint

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We give an overview of some recent developments in semigroup C*-algebras. Contents 1. Introduction 2. C*-algebras generated by left regular representations 3. Examples 3.1. The natural numbers 3.2. Positive cones in totally ordered groups 3.3. Monoids given by presentations 3.4. Examples from rings in general, and number theory in particular 3.5. Finitely generated abelian cancellative semigroups 4. Preliminaries 4.1. Embedding semigroups into groups 4.2. Graph products 4.3. Krull rings 5. C*-algebras attached to inverse semigroups, partial dynamical systems, and groupoids 5.1. Inverse semigroups 5.2. Partial dynamical systems 5.3. Étale groupoids 5.4. The universal groupoid of an inverse semigroup 5.5. Inverse semigroup C*-algebras as groupoid C*-algebras 5.6. C*-algebras of partial dynamical systems as C*-algebras of partial transformation groupoids 5.7. The case of inverse semigroups admitting an idempotent pure partialhomomorphism to a group 6. Amenability and nuclearity 6.1. Groups and groupoids 6.2. Amenability for semigroups 6.3. Comparing reduced C*-algebras for left cancellative semigroups and their left inverse hulls 6.4. C*-algebras generated by semigroups of projections 6.5. The independence condition 6.6. Construction of full semigroup C*-algebras 6.7. Crossed product and groupoid C*-algebra descriptions of reduced semigroup C*-algebras 6.8. Amenability of semigroups in terms of C*-algebras 1 2 XIN LI 6.9. Nuclearity of semigroup C*-algebras and the connection to amenability 68 7. Topological freeness, boundary quotients, and C*-simplicity 69 8. The Toeplitz condition 80 9. Graph products 86 9.1. Constructible right ideals 87 9.2. The independence condition 91 9.3. The Toeplitz condition 94 10. K-theory 97 11. Further developments, outlook, and open questions 99 References 101

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“…K-theory for C * (A + Γ ) and the quotients C * Q (A + Γ ) has been computed in [Iva10] in an ad hoc way, and can also be computed using [CEL13]. Let us explain the computation via the latter route.…”

Section: K-theorymentioning

confidence: 99%

The isomorphism problem for semigroup $C^*$-algebras of right-angled Artin monoids

Eilers¹,

Li²,

Ruiz³

2016

Doc. Math.

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Semigroup C*-algebras for right-angled Artin monoids were introduced and studied by Crisp and Laca. In the paper at hand, we are able to present the complete answer to their question of when such C*-algebras are isomorphic. The answer to this question is presented both in terms of properties of the graph defining the Artin monoids as well as in terms of classification by K-theory, and is obtained using recent results from classification of non-simple C*algebras. Moreover, we are able to answer another natural question: Which of these semigroup C*-algebras for right-angled Artin monoids are isomorphic to graph algebras? We give a complete answer, and note the consequence that many of the C*-algebras under study are semiprojective.

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The $K$-theory of Toeplitz $C^*$-algebras of right-angled Artin groups

Cited by 9 publications

References 15 publications

On the K-Theory of Crossed Products by Automorphic Semigroup Actions

On the K-Theory of Crossed Products by Automorphic Semigroup Actions

Semigroup C*-algebras

The isomorphism problem for semigroup $C^*$-algebras of right-angled Artin monoids

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