On dualities of actions and inclusions

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.

This is a second part of our joint research which is promised in the paper arXiv:1707.07377

References in corpus (1)

• Tracially sequentially-split ${}^{\ast }$-homomorphisms between $C^{\ast }$-algebras II

Cited by in corpus (1)

• Rokhlin property and approximate representability for inclusions of index-finite type