Strong Morita equivalence for inclusions of $C^*$-algebras induced by twisted actions of a countable discrete group

TL;DR

This paper investigates the strong Morita equivalence of inclusions of $C^*$-algebras arising from twisted group actions, establishing conditions under which two such actions are equivalent up to automorphism.

Contribution

It demonstrates that strongly Morita equivalent inclusions from twisted actions imply the actions are also strongly Morita equivalent up to a group automorphism.

Findings

Strong Morita equivalence of inclusions implies equivalence of twisted actions up to automorphism.

Irreducibility of one inclusion plays a key role in the analysis.

The results connect the structure of inclusions with the properties of the underlying group actions.

Abstract

We consider two twisted actions of a countable discrete group on $\sigma$-unital $C^*$-algebras. Then by taking the reduced crossed products, we get two inclusions of $C^*$-algebras. We suppose that they are strongly Morita equivalent as inclusions of $C^*$-algebras. Also, we suppose that one of the inclusions is irreducible, that is, the relative commutant of one of the $\sigma$-unital $C^*$-algebra in the multiplier $C^*$-algebra of the reduced twisted crossed product is trivial. We show that the two actions are then strongly Morita equivalent up to some automorphism of the group.