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October 2020 The Rokhlin property for inclusions of $C^*$-algebras
Hiroyuki Osaka, Tamotsu Teruya
Rocky Mountain J. Math. 50(5): 1785-1792 (October 2020). DOI: 10.1216/rmj.2020.50.1785

Abstract

Let PA be an inclusion of σ-unital C-algebras with a finite index in the sense of Pimsner–Popa. Then we introduce the Rokhlin property for a conditional expectation E from A onto P and show that if A is simple and satisfies any of the property (1)(12) listed in the below, and E has the Rokhlin property, then so does P:

  1. simplicity;

  2. nuclearity;

  3. C-algebras that absorb a given strongly self-absorbing C-algebra 𝒟;

  4. C-algebras of stable rank one;

  5. C-algebras of real rank zero;

  6. C-algebras of nuclear dimension at most n, where n+;

  7. C-algebras of decomposition rank at most n, where n+;

  8. separable simple C-algebras that are stably isomorphic to AF algebras;

  9. separable simple C-algebras that are stably isomorphic to AI algebras;

  10. separable simple C-algebras that are stably isomorphic to AT algebras;

  11. separable simple C-algebras that are stably isomorphic to sequential direct limits of one dimensional NCCW complexes;

  12. separable C-algebras with strict comparison of positive elements.

In particular, when α:G Aut(A) is an action of a finite group G on A with the Rokhlin property in the sense of Nawata, the properties (1)(12) are inherited to the fixed point algebra Aα and the crossed product algebra AαG from A. In the case of a finite index inclusion of unital C-algebras PA if the conditional expectation E:AP has the Rokhlin property in the sense of Izumi, the previous results except (11) are observed in previous works.

Citation

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Hiroyuki Osaka. Tamotsu Teruya. "The Rokhlin property for inclusions of $C^*$-algebras." Rocky Mountain J. Math. 50 (5) 1785 - 1792, October 2020. https://doi.org/10.1216/rmj.2020.50.1785

Information

Received: 28 January 2019; Revised: 9 January 2020; Accepted: 2 March 2020; Published: October 2020
First available in Project Euclid: 5 November 2020

zbMATH: 07274835
MathSciNet: MR4170687
Digital Object Identifier: 10.1216/rmj.2020.50.1785

Subjects:
Primary: 46L55
Secondary: 46L35

Rights: Copyright © 2020 Rocky Mountain Mathematics Consortium

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Vol.50 • No. 5 • October 2020
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