The Rokhlin property vs. Rokhlin dimension 1 on unital Kirchberg algebras

Barlak, Selçuk; Enders, Dominic; Matui, Hiroku; Szabó, Gábor; Winter, Wilhelm

doi:10.4171/jncg/226

Cited by 13 publications

(31 citation statements)

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“…In fact, it is often generic or equivalent to outerness; cf. [HWZ15,BEM+15]. The concept makes direct contact with the striking recent developments in the structure and classification theory of nuclear C * -algebras.…”

Section: Introductionmentioning

confidence: 96%

Rokhlin Dimension for Flows

Hirshberg

Szabó

Winter

et al. 2016

Commun. Math. Phys.

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We introduce a notion of Rokhlin dimension for one parameter automorphism groups of C * -algebras. This generalizes Kishimoto's Rokhlin property for flows, and is analogous to the notion of Rokhlin dimension for actions of the integers and other discrete groups introduced by the authors and Zacharias in previous papers. We show that finite nuclear dimension and absorption of a strongly self-absorbing C * -algebra are preserved under forming crossed products by flows with finite Rokhlin dimension, and that these crossed products are stable. Furthermore, we show that a flow on a commutative C * -algebra arising from a free topological flow has finite Rokhlin dimension, whenever the spectrum is a locally compact metrizable space with finite covering dimension. For flows that are both free and minimal, this has strong consequences for the associated crossed product C * -algebras: Those containing a non-zero projection are classified by the Elliott invariant (for compact manifolds this consists of topological K -theory together with the space of invariant probability measures and a natural pairing given by the Ruelle-Sullivan map).

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Section: Introductionmentioning

confidence: 96%

Rokhlin Dimension for Flows

Hirshberg

Szabó

Winter

et al. 2016

Commun. Math. Phys.

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“…In this section, we show that pointwise outer actions of amenable, residually finite groups on Kirchberg algebras have Rokhlin dimension at most one in the sense of [24,77]. Note that this has been observed for symmetries in [2] and for finite groups in [18]. See also [45,46] for results in this direction in the context of finite C * -algebras.…”

Section: Finiteness Of Rokhlin Dimensionmentioning

confidence: 64%

“…In particular, (D, α) admits unital and approximately central and equivariant embeddings into (D ⊗ O ∞ , id D ⊗γ). It then follows from [76, Theorem 2.7] that id D ⊗γ ≃ scc α ⊗ id D ⊗γ.The last part of the claim follows from Proposition 4.5(2) and the argument analogous to Proposition 4.4(2).…”

mentioning

confidence: 78%

“…For the second part, let α (i) : G O 2 be two pointwise outer, equivariantly O 2 -absorbing actions for i = 1, 2. Applying the first part twice, we obtain α (1) ≃ scc α (1) ⊗ α (2) ≃ scc α (2) . This finishes the proof.…”

Section: Uniqueness Of O 2 -Absorbing Outer Actionsmentioning

confidence: 96%

“…Let us recall a useful well-known fact, which was implicitely used for example in [2] and [56]. (2) there exist two equivariant c.p.c.…”

Section: Finiteness Of Rokhlin Dimensionmentioning

confidence: 99%

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Equivariant Kirchberg–Phillips-Type Absorption for Amenable Group Actions

Szabó

2018

Commun. Math. Phys.

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We show an equivariant Kirchberg-Phillips-type absorption theorem for pointwise outer actions of discrete amenable groups on Kirchberg algebras with respect to natural model actions on the Cuntz algebras O∞ and O2. This generalizes results known for finite groups and poly-Z groups. The model actions are shown to be determined, up to strong cocycle conjugacy, by natural abstract properties, which are verified for some examples of actions arising from tensorial shifts. We also show the following homotopy rigidity result, which may be understood as a precursor to a general Kirchberg-Phillips-type classification theory: If two outer actions of an amenable group on a unital Kirchberg algebra are equivariantly homotopy equivalent, then they are conjugate. This marks the first C * -dynamical classification result up to cocycle conjugacy that is applicable to actions of all amenable groups.2010 Mathematics Subject Classification. 46L55 (primary); 46L05, 19K35 (secondary).

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Nuclear dimension and $$\mathcal Z$$ Z -stability

2015

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Simple, separable, unital, monotracial and nuclear C * -algebras are shown to have finite nuclear dimension whenever they absorb the Jiang-Su algebra Z tensorially. This completes the proof of the Toms-Winter conjecture in the unique trace case.The structure theory of simple nuclear C * -algebras is currently undergoing revolutionary progress, driven by the discovery of regularity properties of various flavours: topological, functional analytic and algebraic. Despite the diverse nature of these regularity properties, they are all satisfied by those classes of C * -algebras which have been successfully classified by K -theoretic data, and they all fail spectacularly for the "exotic" algebras in [30,40] which provide counterexamples to Elliott's classification conjecture. The observation that

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The Rokhlin property vs. Rokhlin dimension 1 on unital Kirchberg algebras

Cited by 13 publications

References 13 publications

Rokhlin Dimension for Flows

Rokhlin Dimension for Flows

Equivariant Kirchberg–Phillips-Type Absorption for Amenable Group Actions

Nuclear dimension and $$\mathcal Z$$ Z -stability

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