Quasi-free actions of finite groups on the Cuntz algebra $\mathcal{O}_\infty$

Goldstein, Pavle; Izumi, Masaki

doi:10.2748/tmj/1325886288

Cited by 19 publications

(22 citation statements)

References 17 publications

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“…The reader is referred to [36] and [39] for the classification of Kirchberg algebras, and to Introduction of the first part [22] for a brief account of the classification of group actions on classifiable C * -algebras. Here we just give an incomplete list [1], [14], [17], [18], [20], [24], [25], [26], [27], [28], [30], [31], [34], [35] of classification results on discrete amenable group actions on classifiable C * -algebras.…”

Section: Introductionmentioning

confidence: 99%

Poly-ℤ Group Actions on Kirchberg Algebras I

Izumi

Matui

2019

International Mathematics Research Notices

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This is the second part of our serial work on the classification of poly-Z group actions on Kirchberg algebras. Based on technical results obtained in our previous work, we completely reduce the problem to the classification of continuous fields of Kirchberg algebras over the classifying spaces. As an application, we determine the number of cocycle conjugacy classes of outer Z n -actions on the Cuntz algebras.

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Section: Introductionmentioning

confidence: 99%

Poly-ℤ Group Actions on Kirchberg Algebras I

Izumi

Matui

2019

International Mathematics Research Notices

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“…This technique will lead us to prove a special cohomology vanishing result for specific kinds of cocycles, which will form a useful technical tool for our purposes. As an application of this technical tool, we prove an existence result for G-equivariant * -homomorphisms from certain model actions into certain saturated G-C * -dynamical systems, in part drawing inspiration from [20]. This approach is in line with the philosophy that the concept of saturation (in its various forms), just like ultrapowers, allows one to carry over certain techniques to solve approximation problems that would otherwise be inaccessible outside of a finitary situation.…”

Section: Introductionmentioning

confidence: 84%

“…It is even unclear whether it is redundant if one assumes that α is KK G -equivalent to the trivial Gaction on D. This latter condition is known to hold if α is a quasi-free action on O ∞ ; see [65,Section 4]. Indeed, the main result of [20] shows that for finite groups, any faithful and quasi-free action on O ∞ is conjugate to the model action in Example 3.6. However, the proof uses certain techniquess that are not readily available for infinite groups.…”

Section: Remark 412mentioning

confidence: 99%

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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“…Note that Izumi also showed that there exist uncountably many mutually non-conjugate outer actions of Z 2 on O 2 . Also, Goldstein and Izumi obtained an equivariant Kirchberg-Phillips type result for finite group actions on O ∞ in [17].…”

Section: Introductionmentioning

confidence: 99%

Rohlin Actions of Finite Groups on the Razak–Jacelon Algebra

Nawata

2020

International Mathematics Research Notices

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Let A be a simple separable nuclear C * -algebra with a unique tracial state and no unbounded traces, and let α be a strongly outer action of a finite group G on A. In this paper, we show that α ⊗ id on A ⊗ W has the Rohlin property, where W is the Razak-Jacelon algebra. Combing this result with the recent classification results and our previous result, we see that such actions are unique up to conjugacy.2010 Mathematics Subject Classification. Primary 46L55, Secondary 46L35; 46L40.

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Quasi-free actions of finite groups on the Cuntz algebra $\mathcal{O}_\infty$

Abstract: We show that any faithful quasi-free actions of a finite group on the Cuntz algebra O ∞ are mutually conjugate, and that they are asymptotically representable.

Cited by 19 publications

References 17 publications

Poly-ℤ Group Actions on Kirchberg Algebras I

Poly-ℤ Group Actions on Kirchberg Algebras I

Equivariant Kirchberg–Phillips-Type Absorption for Amenable Group Actions

Rohlin Actions of Finite Groups on the Razak–Jacelon Algebra

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