Crossed products of nuclear C⁎-algebras and their traces

Rainone, Timothy; Schafhauser, Christopher

doi:10.1016/j.aim.2019.01.045

Cited by 6 publications

(6 citation statements)

References 46 publications

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“…In the amenable case we recover quasidiagonality. That a certain coboundary condition is equivalent to stable finiteness is reminiscent of the work of Pimsner [30] and Brown [8] and also appears in noncommutative C * -systems [10,31,34]. In Theorem 7.3 we establish that, again for minimal groupoids with totally disconnected unit space, if every element of S(G) is properly infinite, then C * (G) is purely infinite; moreover these conditions are equivalent if S(G) is almost unperforated.…”

Section: Introductionmentioning

confidence: 68%

A dichotomy for groupoid -algebras

Rainone

Sims

2018

Ergod. Th. Dynam. Sys.

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We study the finite versus infinite nature of C * -algebras arising frométale groupoids. For an ample groupoid G, we relate infiniteness of the reduced C * -algebra of G to notions of paradoxicality of a K-theoretic flavor. We construct a pre-ordered abelian monoid S(G) which generalizes the type semigroup introduced by Rørdam and Sierakowski for totally disconnected discrete transformation groups. This monoid reflects the finite/infinite nature of the reduced groupoid C * -algebra of G. If G is ample, minimal, and topologically principal, and S(G) is almost unperforated we obtain a dichotomy between stable finiteness and pure infiniteness for the reduced C * -algebra of G.

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Section: Introductionmentioning

confidence: 68%

A dichotomy for groupoid -algebras

Rainone

Sims

2018

Ergod. Th. Dynam. Sys.

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“…We now state a general result used to prove Theorem C. The equivalence (2)⇔(3) of Proposition 2.9 have been considered previously, see for example [13,Proposition 3.1]. For results related to (3)⇒(2) when A is unital we refer to [39,Theorem 4.2]. Proposition 2.9.…”

Section: Lemma 26 ([8]mentioning

confidence: 99%

Unbounded quasitraces, stable finiteness and pure infiniteness

Pask¹,

Sierakowski²,

Sims³

2017

Preprint

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We prove that if A is a σ-unital exact C * -algebra of real rank zero, then every state on K 0 (A) is induced by a 2-quasitrace on A. This yields a generalisation of Rainone's work on pure infiniteness and stable finiteness of crossed products to the non-unital case. It also applies to k-graph algebras associated to row-finite k-graphs with no sources. We show that for any k-graph whose C * -algebra is unital and simple, either every twisted C * -algebra associated to that k-graph is stably finite, or every twisted C *algebra associated to that k-graph is purely infinite. Finally we provide sufficient and necessary conditions for a unital simple k-graph algebra to be purely infinite in terms of the underlying k-graph. Contents 8 3. Dichotomy for k-graph C * -algebras and the proof of Theorem D 14 4. Stable finiteness of twisted k-graph algebras and the proof of Theorem E 20 5. Purely infinite twisted k-graph algebras and the proof of Theorem F 23 6. The semigroup S(Λ) 28 References 36

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“…In fact, under the assumptions of Theorem 5.10, one can conclude that the inclusion of K 0 (A α ) into K 0 (A) is positively existential. Here we regard K 0 -group as structures in the language of dimension groups (G, +, u) endowed with domains of quantifications to be interpreted as the subsets {x ∈ G : − nu ≤ x ≤ nu} for n ∈ N. This corresponds to the notion of ultrapower of dimension groups considered in [46]. G is a compact quantum group, and (B, β) is a G-C*-algebra.…”

Section: The Rokhlin Propertymentioning

confidence: 99%

Model theory and Rokhlin dimension for compact quantum group actions

Gardella

Kalantar

Lupini

2019

J. Noncommut. Geom.

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We show that, for a given compact or discrete quantum group G, the class of actions of G on C*-algebras is first-order axiomatizable in the logic for metric structures. As an application, we extend the notion of Rokhlin property for G-C*-algebra, introduced by Barlak, Szabó, and Voigt in the case when G is second countable and coexact, to an arbitrary compact quantum group G. All the the preservations and rigidity results for Rokhlin actions of second countable coexact compact quantum groups obtained by Barlak, Szabó, and Voigt are shown to hold in this general context. As a further application, we extend the notion of equivariant order zero dimension for equivariant *-homomorphisms, introduced in the classical setting by the first and third authors, to actions of compact quantum groups. This allows us to define the Rokhlin dimension of an action of a compact quantum group on a C*-algebra, recovering the Rokhlin property as Rokhlin dimension zero. We conclude by establishing a preservation result for finite nuclear dimension and finite decomposition rank when passing to fixed point algebras and crossed products by compact quantum group actions with finite Rokhlin dimension.

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Crossed products of nuclear C⁎-algebras and their traces

Cited by 6 publications

References 46 publications

A dichotomy for groupoid -algebras

A dichotomy for groupoid -algebras

Unbounded quasitraces, stable finiteness and pure infiniteness

Model theory and Rokhlin dimension for compact quantum group actions

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