Strongly outer actions of amenable groups on<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mi mathvariant="script">Z</mml:mi></mml:math>-stable nuclear C*-algebras

Gardella, Eusebio; Hirshberg, Ilan; Vaccaro, Andrea

doi:10.1016/j.matpur.2022.04.003

Cited by 8 publications

(3 citation statements)

References 66 publications

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“…We are indebted to Stefaan Vaes for pointing this out to us in the context of the above question and for outlining a sketch of proof on how to correct this, which became a sort of blueprint for the main result of the fourth section. In contemporary research by C * -algebraists, the aforementioned results by Sutherland-Takesaki are still used to provide a partial answer to the above question, for example, in [15]. In light of the previous discussion, the present article aims to give a self-containedand dare we say also relatively elementary -approach to answer this question instead.…”

Section: Question Let G Be a Countable Amenable Group And M A Separab...mentioning

confidence: 91%

“…If one is concerned with C * -algebraic considerations related to the equivariant Jiang-Su stability problem (see [47, Conjecture A]), the current methods always find a way to exploit Oceanu's aforementioned theorem in one form or another, usually involving to some degree Matui-Sato's property (SI) technique [27,28,29,42]. Looking at the state-ofthe-art at present [15,54], the key difficulties arise from pushing these methods to the case where a group action G A on a C * -algebra induces a complicated G-action on the traces of A. In particular, it is generally insufficient for such considerations to only understand G-actions on R, but one rather needs to have control over G-actions on more general tracial von Neumann algebras.…”

Section: Introductionmentioning

confidence: 99%

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Dynamical McDuff-Type Properties for Group Actions on Von Neumann Algebras

Szabó,

Wouters

2024

J. Inst. Math. Jussieu

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We consider the notion of strong self-absorption for continuous actions of locally compact groups on the hyperfinite II $_1$ factor and characterize when such an action is tensorially absorbed by another given action on any separably acting von Neumann algebra. This extends the well-known McDuff property for von Neumann algebras and is analogous to the core theorems around strongly self-absorbing C $^*$ -dynamics. Given a countable discrete group G and an amenable action $G\curvearrowright M$ on any separably acting semifinite von Neumann algebra, we establish a type of measurable local-to-global principle: If a given strongly self-absorbing G-action is suitably absorbed at the level of each fibre in the direct integral decomposition of M, then it is tensorially absorbed by the action on M. As a direct application of Ocneanu’s theorem, we deduce that if M has the McDuff property, then every amenable G-action on M has the equivariant McDuff property, regardless whether M is assumed to be injective or not. By employing Tomita–Takesaki theory, we can extend the latter result to the general case, where M is not assumed to be semifinite.

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Section: Question Let G Be a Countable Amenable Group And M A Separab...mentioning

confidence: 91%

Section: Introductionmentioning

confidence: 99%

Dynamical McDuff-Type Properties for Group Actions on Von Neumann Algebras

Szabó,

Wouters

2024

J. Inst. Math. Jussieu

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“…We say that α is strongly outer if α is an outer action on π τA (A) ′′ . (We refer the reader to [6] and [25] for the definition of strongly outerness for more general settings. )…”

Section: Preliminariesmentioning

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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Tracially amenable actions and purely infinite crossed products

Gardella,

Geffen,

Kranz

et al. 2024

Math. Ann.

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We introduce the notion of tracial amenability for actions of discrete groups on unital, tracial C$$^*$$ ∗ -algebras, as a weakening of amenability where all the relevant approximations are done in the uniform trace norm. We characterize tracial amenability with various equivalent conditions, including topological amenability of the induced action on the trace space. Our main result concerns the structure of crossed products: for groups containing the free group $$F_2$$ F 2 , we show that outer, tracially amenable actions on simple, unital, $$\mathcal {Z}$$ Z -stable C$$^*$$ ∗ -algebras always have purely infinite crossed products. Finally, we give concrete examples of tracially amenable actions of free groups on simple, unital AF-algebras.

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Strongly outer actions of amenable groups onZ-stable nuclear C*-algebras

Cited by 8 publications

References 66 publications

Dynamical McDuff-Type Properties for Group Actions on Von Neumann Algebras

Dynamical McDuff-Type Properties for Group Actions on Von Neumann Algebras

Rohlin Actions of Finite Groups on the Razak–Jacelon Algebra

Tracially amenable actions and purely infinite crossed products

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