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

TL;DR

This paper introduces a new Rokhlin property for inclusions of unital C*-algebras, including those without projections, and explores their duality and implications for algebraic properties.

Contribution

It defines a novel Rokhlin property for unital C*-algebra inclusions and establishes a duality with approximate representability, impacting structural properties.

Findings

Proves permanence of tracial Z-absorbingness

Shows preservation of strict comparison property

Establishes duality between Rokhlin property and approximate representability

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.