On dualities of actions and inclusions

TL;DR

This paper establishes duality results linking Rokhlin properties and approximate representability in inclusions of $C^*$-algebras with finite Watatani index, extending known finite abelian group action results.

Contribution

It proves duality theorems for Rokhlin and tracial Rokhlin properties in index-finite inclusions, and offers a new proof of Phillips' theorem using a novel conceptual framework.

Findings

Duality between Rokhlin property and approximate representability.

Duality between tracial Rokhlin property and tracial approximate representability.

New proof of Phillips' theorem on tracial Rokhlin actions.

Abstract

Following the results known in the case of a finite abelian group action on $C\sp*$-algebras we prove the following two theorems; 1. an inclusion $P\subset A$ of (Watatani) index-finite type has the Rokhlin property (is approximately representable) if and only if the dual inclusion is approximately representable (has the Rokhlin property). 2. an inclusion $P\subset A$ of (Watatani) index-finite type has the tracial Rokhlin property (is tracially approximately representable) if and only if the dual inclusion is tracially approximately representable (has the tracial Rokhlin property). Moreover, we provide an alternate proof of Phillips' theorem about the relations between tracial Rokhlin action and tracially approximate representable dual action using a new conceptual framework suggested by authors.