The Classification of Rokhlin Flows on $$\mathrm {C}^{*}$$-algebras

Szabó, Gábor

doi:10.1007/s00220-020-03812-2

Cited by 9 publications

(8 citation statements)

References 80 publications

(144 reference statements)

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“…By [73,Theorem 5.12], it suffices to prove that every strongly outer action α : Z A has the Rokhlin property. If we apply Corollary 5.24 to the special case Γ = Z, it follows that every such action α tensorially absorbs some automorphism on Q with the Rokhlin property, and will therefore itself have the Rokhlin property.…”

Section: Applicationsmentioning

confidence: 99%

Equivariant property (SI) revisited

Szabó

2021

Analysis & PDE

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We revisit Matui-Sato's notion of property (SI) for C *algebras and C * -dynamics. More specifically, we generalize the known framework to the case of C * -algebras with possibly unbounded traces. The novelty of this approach lies in the equivariant context, where none of the previous work allows one to (directly) apply such methods to actions of amenable groups on highly non-unital C * -algebras, in particular to establish equivariant Jiang-Su stability. Our main result is an extension of an observation by Sato: For any countable amenable group Γ and any non-elementary separable simple nuclear C * -algebra A with strict comparison, every Γ-action on A has equivariant property (SI). A more general statement involving relative property (SI) for inclusions into ultraproducts is proved as well. As a consequence we show that if A also has finitely many rays of extremal traces, then every Γ-action on A is equivariantly Jiang-Su stable. We moreover provide applications of the main result to the context of strongly outer actions, such as a generalization of Nawata's classification of strongly outer automorphisms on the (stabilized) Razak-Jacelon algebra. Contents 12 4. Equivariant property (SI) 19 5. Applications 21 References 32

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Section: Applicationsmentioning

confidence: 99%

Equivariant property (SI) revisited

Szabó

2021

Analysis & PDE

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“…The Rokhlin property for flows was introduced by Kishimoto in [59], who provided evidence why one should expect that these can be classified up to cocycle conjugacy [65,6]. This was confirmed in my recent work [95], which was in part inspired by [71].…”

Section: Introductionmentioning

confidence: 79%

On a categorical framework for classifying C*-dynamics up to cocycle conjugacy

Szabo

2019

Preprint

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We provide the rigorous foundations for a categorical approach to the classification of C * -dynamics up to cocycle conjugacy. Given a locally compact group G, we consider a category of (twisted) G-C * -algebras, where morphisms between two objects are allowed to be equivariant maps or exterior equivalences, which leads to the concept of so-called cocycle morphisms. An isomorphism in this category is precisely a cocycle conjugacy in the known sense. We show that this category allows sequential inductive limits, and that some known functors on the usual category of G-C * -algebras extend. After observing that this setup allows a natural notion of (approximate) unitary equivalence, the main aim of the paper is to generalize the fundamental intertwining results commonly employed in the Elliott program for classifying C *algebras. This reduces a given classification problem for C * -dynamics to the prevalence of certain uniqueness and existence theorems, and may provide a useful alternative to the Evans-Kishimoto intertwining argument in future research.

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“…Although the theorem above is not too far off from being a very special case of [79] for ordinary flows, this result is entirely new for k ≥ 2, and is in fact the first classification result for R k -actions on C * -algebras up to cocycle conjugacy.…”

Section: Introductionmentioning

confidence: 79%

“…At this moment it seems unclear whether or not to expect a similarly rigid situation for Rokhlin R k -actions on general Kirchberg algebras, as is the case for k = 1 [79]. In general, in order to implement a more general classification of this sort, it would require a technique for both constructing and manipulating cocycles for R k -actions (where k ≥ 2) with the help of the Rokhlin property, which may potentially be much more complicated than for k = 1.…”

Section: Introductionmentioning

confidence: 99%

Rokhlin dimension: absorption of model actions

Szabó

2019

Analysis & PDE

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In this paper, we establish a connection between Rokhlin dimension and the absorption of certain model actions on strongly selfabsorbing C * -algebras. Namely, as to be made precise in the paper, let G be a well-behaved locally compact group. If D is a strongly selfabsorbing C * -algebra, and α : G A is an action on a separable, Dabsorbing C * -algebra that has finite Rokhlin dimension with commuting towers, then α tensorially absorbs every semi-strongly self-absorbing Gactions on D. In particular, this is the case when α satisfies any version of what is called the Rokhlin property, such as for G = R or G = Z k . This contains several existing results of similar nature as special cases. We will in fact prove a more general version of this theorem, which is intended for use in subsequent work. We will then discuss some nontrivial applications. Most notably it is shown that for any k ≥ 1 and on any strongly self-absorbing Kirchberg algebra, there exists a unique R k -action having finite Rokhlin dimension with commuting towers up to (very strong) cocycle conjugacy.

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The Classification of Rokhlin Flows on $$\mathrm {C}^{*}$$-algebras

Cited by 9 publications

References 80 publications

Equivariant property (SI) revisited

Equivariant property (SI) revisited

On a categorical framework for classifying C*-dynamics up to cocycle conjugacy

Rokhlin dimension: absorption of model actions

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