K-theory of certain purely infinite crossed products

Elliott, George A.; Sierakowski, Adam

doi:10.1016/j.jmaa.2016.05.001

Cited by 5 publications

(13 citation statements)

References 18 publications

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“…In this case, each Γ-Cantor system can be taken to be free. This follows from the proof of Theorem 5.1 in [7], the existence of a minimal Cantor F n -system whose K 1 -group is isomorphic to Z ⊕∞ , and the Pimsner-Voiculescu exact sequence.…”

Section: More General Constructions Of Amenable Minimal Cantormentioning

confidence: 86%

“…As examples, we show that the diagonal actions of the boundary actions and the products of odometer transformations are classified in terms of continuous orbit equivalence by using a C * -algebraic technique. A recent work of G. A. Elliott and A. Sierakowski [7] gives us an example of amenable minimal Cantor F n -systems which are distinguished by K-theory. They construct an amenable minimal free Cantor F n -system whose K 0 -group vanishes.…”

Section: 1mentioning

confidence: 99%

“…Their construction is based on the idea developed in the paper [25]. Our strategy is different from that of [7]. We construct amenable minimal Cantor F n -systems from the diagonal actions.…”

Section: 1mentioning

confidence: 99%

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Amenable minimal Cantor systems of free groups arising from diagonal actions

Suzuki

2014

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Abstract. We study amenable minimal Cantor systems of free groups arising from the diagonal actions of the boundary actions and certain Cantor systems. It is shown that every virtually free group admits continuously many amenable minimal Cantor systems whose crossed products are mutually non-isomorphic Kirchberg algebras in the UCT class (with explicitly determined K-theory). The technique developed in our study also enables us to compute the K-theory of certain amenable minimal Cantor systems. We apply it to the diagonal actions of the boundary actions and the products of the odometer transformations, and determine their K-theory. Then we classify them in terms of the topological full groups, continuous orbit equivalence, strong orbit equivalence, and the crossed products.

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Section: More General Constructions Of Amenable Minimal Cantormentioning

confidence: 86%

Section: 1mentioning

confidence: 99%

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Amenable minimal Cantor systems of free groups arising from diagonal actions

Suzuki

2014

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…First, note that the uniform Roe algebra of a bounded geometry space

X

contains a copy of

ℓ^{\infty} false(X false)

as diagonal matrices. The following theorem from Section 5 extends work of Elliott and Sierakowski [, Section 5] on the case when

X

is a free group. Theorem Let

X

be a metric space with bounded geometry and let

δ : ℓ^{\infty} false(X false) \to C_{u}^{*} false(X false)

be the inclusion of the diagonal.…”

Section: Introductionmentioning

confidence: 58%

“…In particular, when

X

is the fundamental group of a surface of genus

two

this answers a question of Elliott and Sierakowski: they asked in [, Question 5.2] whether

K_{0} false(C_{u}^{*} false| G false| false)

is always zero for non‐amenable groups, and the above shows the answer is ‘no’. The proof of the above theorem uses higher index theory to construct interesting classes in

K_{0} false(C_{u}^{*} (X) false)

.…”

Section: Introductionmentioning

confidence: 96%

Low‐dimensional properties of uniform Roe algebras

Willett

2018

J. London Math. Soc.

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Abstract. The goal of this paper is to study when uniform Roe algebras have certain C * -algebraic properties in terms of the underlying space: in particular, we study properties like having stable rank one or real rank zero that are thought of as low dimensional, and connect these to low dimensionality of the underlying space in the sense of the asymptotic dimension of Gromov. Some of these results (for example, on stable rank one, cancellation, strong quasidiagonality, and finite decomposition rank) give definitive characterizations, while others (on real rank zero) are only partial and leave a lot open.We also establish results about K-theory, showing that all K0-classes come from the inclusion of the canonical Cartan in low dimensional cases, but not in general; in particular, our K-theoretic results answer a question of Elliott and Sierakowski about vanishing of K0 groups for uniform Roe algebras of non-amenable groups. Along the way, we extend some results about paradoxicality, proper infiniteness of projections in uniform Roe algebras, and supramenability from groups to general metric spaces. These are ingredients needed for our K-theoretic computations, but we also use them to give new characterizations of supramenability for metric spaces. Mathematics Subject Classification (2010): 20F69, 46L05, 46L80, 55M10

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