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The tracial Rokhlin property for an inclusion of unital $C^*$-algebras

Hyun Ho Lee, Hiroyuki Osaka

Abstract

We introduce and study a notion of Rokhlin property for an inclusion of unital $C^*$-algebras which could have no projections like the Jiang-Su algebra. We also introduce a notion of approximate representability and show a duality between them. We demonstrate the importance of these notions by showing the permanence of the tracial $\mathcal{Z}$-absorbingness and the strict comparison property.

The tracial Rokhlin property for an inclusion of unital $C^*$-algebras

Abstract

We introduce and study a notion of Rokhlin property for an inclusion of unital -algebras which could have no projections like the Jiang-Su algebra. We also introduce a notion of approximate representability and show a duality between them. We demonstrate the importance of these notions by showing the permanence of the tracial -absorbingness and the strict comparison property.

Paper Structure

This paper contains 7 sections, 27 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and notations
  3. The generalized tracial Rokhlin property for an inclusion of unital $C\sp*$-algebras of index-finite type
  4. The generalized tracial approximate representability for an inclusion of unital $C\sp*$-algebras of index-finite type
  5. Dualities between Rokhlin property and approximate representability
  6. Applications
  7. Acknowledgements

Key Result

Theorem 2.3

Izumi:inclusion Let $P\subset A$ be an inclusion of unital $C\sp*$-algebras of index-finite type. Then Moreover, if $A=\oplus A_i$ then each $A_i$ is of the form $Az$ where $z$ is the projection in $Z(A)$ the center of $A$.

Theorems & Definitions (64)

  • Definition 2.1: Watatani
  • Definition 2.2
  • Theorem 2.3
  • Definition 2.4
  • Theorem 2.5: Winter and Zacharias
  • Lemma 2.6
  • proof
  • Proposition 2.7
  • proof
  • Lemma 2.8
  • ...and 54 more