Complete Descriptions of Intermediate Operator Algebras by Intermediate Extensions of Dynamical Systems

Suzuki, Yuhei

doi:10.1007/s00220-019-03436-1

Cited by 22 publications

(42 citation statements)

References 71 publications

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“…Intermediate C * -algebras, i.e. C * -algebras B of the form C * r ( ) ⊆ B ⊆ r A, have recently gained some particular attention, for instance in the work of Suzuki [15,16] in connection to problems of minimal ambient nuclear C * -algebras as well as maximal injective von Neumann subalgebras.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

On simplicity of intermediate -algebras

Amrutam¹,

Kalantar²

2019

Ergod. Th. Dynam. Sys.

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We prove simplicity of all intermediate $C^{\ast }$ -algebras $C_{r}^{\ast }(\unicode[STIX]{x1D6E4})\subseteq {\mathcal{B}}\subseteq \unicode[STIX]{x1D6E4}\ltimes _{r}C(X)$ in the case of minimal actions of $C^{\ast }$ -simple groups $\unicode[STIX]{x1D6E4}$ on compact spaces $X$ . For this, we use the notion of stationary states, recently introduced by Hartman and Kalantar [Stationary $C^{\ast }$ -dynamical systems. Preprint, 2017, arXiv:1712.10133]. We show that the Powers’ averaging property holds for the reduced crossed product $\unicode[STIX]{x1D6E4}\ltimes _{r}{\mathcal{A}}$ for any action $\unicode[STIX]{x1D6E4}\curvearrowright {\mathcal{A}}$ of a $C^{\ast }$ -simple group $\unicode[STIX]{x1D6E4}$ on a unital $C^{\ast }$ -algebra ${\mathcal{A}}$ , and use it to prove a one-to-one correspondence between stationary states on ${\mathcal{A}}$ and those on $\unicode[STIX]{x1D6E4}\ltimes _{r}{\mathcal{A}}$ .

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Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

On simplicity of intermediate -algebras

Amrutam¹,

Kalantar²

2019

Ergod. Th. Dynam. Sys.

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“…Then the inclusion Λ O 2 ⊂ M extends to the Λ-equivariant inclusion A ⊂ M¯ β Γ. Since ( Λ O 2 ) σ = C (by lemma 2.1), by corollary 3.4 of [33], we have A θ ⊂ L(Γ). Moreover, when Γ has the AP, thanks to proposition 3.4 of [31], we further obtain A θ = C * r (Γ).…”

Section: Proofs Constructions and Remarksmentioning

confidence: 91%

“…We refer the reader to the book [29] for fundamental facts and backgrounds on this subject. The key ingredients of our constructions (besides well-known deep results) are 'amenable' actions of non-amenable groups on Kirchberg algebras recently obtained in [32,33].…”

Section: Of Isomorphs Of O 2 Whose Intersection Does Not Have the Opementioning

confidence: 99%

“…Let Γ be a countable free group of infinite rank. In the proof of theorem 5.1 of [33], we have constructed an action α : Γ O ∞ which contains a unital amenable Γ-C * -subalgebra in the sense of definition 4.3.1 in [2]. By replacing α by the diagonal action of α and the trivial Γ-action on O ∞ (cf.…”

Section: Proof Of Proposition Thanks To Kirchberg's O ∞ -Absorption mentioning

confidence: 99%

“…(See the second paragraph of the proof of theorem 5.1 in [33] for details of computation.) Let ζ : Γ Γ O ∞ denote the tensor shift action.…”

Section: Proof Of Proposition Thanks To Kirchberg's O ∞ -Absorption mentioning

confidence: 99%

See 2 more Smart Citations

On pathological properties of fixed point algebras in Kirchberg algebras

Suzuki

2019

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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We investigate how the fixed point algebra of a C * -dynamical system can differ from the underlying C * -algebra. For any exact group Γ and any infinite group Λ, we construct an outer action of Λ on the Cuntz algebra O 2 whose fixed point algebra is almost equal to the reduced group C * -algebra C * r (Γ). Moreover, we show that every infinite group admits outer actions on all Kirchberg algebras whose fixed point algebras fail the completely bounded approximation property.2000 Mathematics Subject Classification. Primary 22D25, 46L55, Secondary 46L05.

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On characterizations of amenable $$\hbox {C}^*$$-dynamical systems and new examples

Ozawa

Suzuki

2021

Sel. Math. New Ser.

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We show the equivalence of several amenability type conditions for $$\hbox {C}^*$$ C ∗ -dynamical systems. As an application, based on the Pimsner–Meyer construction, we give a new method to produce amenable actions on simple $$\hbox {C}^*$$ C ∗ -algebras.

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Complete Descriptions of Intermediate Operator Algebras by Intermediate Extensions of Dynamical Systems

Cited by 22 publications

References 71 publications

On simplicity of intermediate -algebras

On simplicity of intermediate -algebras

On pathological properties of fixed point algebras in Kirchberg algebras

On characterizations of amenable $$\hbox {C}^*$$-dynamical systems and new examples

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