Strongly outer actions of amenable groups on $\mathcal{Z}$-stable $C^*$-algebras

Gardella, Eusebio; Hirshberg, Ilan

doi:10.48550/arxiv.1811.00447

Cited by 6 publications

(12 citation statements)

References 24 publications

(71 reference statements)

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“…We say that α is strongly outer if αg is not an inner automorphism of π τA (A) ′′ for any g ∈ Γ \ {ι}. (See, for example, [14] for the definition of strongly outerness in more general settings.) It is known that if α is a strongly outer action of Γ on a simple monotracial C * -algebra A, then A ⋊ α Γ is simple and monotracial.…”

Section: Preliminariesmentioning

confidence: 99%

Equivariant Kirchberg-Phillips type absorption for the Razak-Jacelon algebra

Nawata¹

2021

Preprint

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Let A and B be simple separable nuclear monotracial C * -algebras, and let α and β be strongly outer actions of a countable discrete amenable group Γ on A and B, respectively. In this paper, we show that α ⊗ id W on A ⊗ W and β ⊗ id W on B ⊗ W are cocycle conjugate where W is the Razak-Jacelon algebra. Also, we characterize such actions by using the fixed point subalgebras of Kirchberg's central sequence C * -algebras.

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Section: Preliminariesmentioning

confidence: 99%

Equivariant Kirchberg-Phillips type absorption for the Razak-Jacelon algebra

Nawata¹

2021

Preprint

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“…To prove Condition (4), by (3) and that x ∈ F we have for an equivalent definition.) Definition 3.7 (see [15]). Let α : G → Aut(A) be an action of a finite group G on a simple unital C*-algebra A.…”

Section: The Weak Tracial Rokhlin Propertymentioning

confidence: 99%

The Weak Tracial Rokhlin Property for Finite Group Actions on Simple C*-Algebras

Forough

Golestani

2020

Doc. Math.

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We develop the concept of weak tracial Rokhlin property for finite group actions on simple (not necessarily unital) C*-algebras and study its properties systematically. In particular, we show that this property is stable under restriction to invariant hereditary C*algebras, minimal tensor products, and direct limits of actions. Some of these results are new even in the unital case and answer open questions asked by N. C. Phillips in full generality. We present several examples of finite group actions with the weak tracial Rokhlin property on simple stably projectionless C*-algebras. We prove that if α : G → Aut(A) is an action of a finite group G on a simple C*algebra A with tracial rank zero and α has the weak tracial Rokhlin property, then the crossed product A ⋊ α G and the fixed point algebra A α are simple with tracial rank zero. This extends a result of N. C. Phillips to the nonunital case. We use the machinery of Cuntz subequivalence to work in this nonunital setting.

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“…Those form a central sequence of Rokhlin towers in C * r (K), and thus the action α on C * r (K) has the Rokhlin property. We recall the following characterization of the existence of a unital homomorphism from a dimension drop algebra from [25, Proposition 5.1] (see also [8,Theorem 4.8]). Proposition 3.4.…”

Section: Nearly Approximately Inner Automorphismsmentioning

confidence: 99%

Rokhlin-type properties, approximate innerness and Z-stability

Hirshberg¹

2020

Preprint

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We establish three results concerning connections between actions on separable C * -algebras with finite Rokhlin dimension with commuting towers and absorption of the Jiang-Su algebra Z. For actions of residually finite groups or of the reals, we show that if the action of any nontrivial group element is approximately inner then the C * -algebra acted upon is Z-stable. Without the assumption on approximate innerness, we show that the crossed product has good divisibility properties under mild assumptions. For actions of a single automorphism which have the Rokhlin property, we show that a condition which is strictly weaker than requiring that some power of the automorphism is approximately inner is sufficient to obtain that the crossed product absorbs Z even when the original algebra is not.

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Strongly outer actions of amenable groups on $\mathcal{Z}$-stable $C^*$-algebras

Cited by 6 publications

References 24 publications

Equivariant Kirchberg-Phillips type absorption for the Razak-Jacelon algebra

Equivariant Kirchberg-Phillips type absorption for the Razak-Jacelon algebra

The Weak Tracial Rokhlin Property for Finite Group Actions on Simple C*-Algebras

Rokhlin-type properties, approximate innerness and Z-stability

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