On dualities of actions and inclusions
Hyun Ho Lee, Hiroyuki Osaka

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 -algebras we prove the following two theorems; 1. an inclusion 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 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.
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Taxonomy
TopicsAdvanced Operator Algebra Research · Advanced Topics in Algebra · Advanced Banach Space Theory
