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Strong Morita equivalence for inclusions of $C^*$-algebras induced by twisted actions of a countable discrete group (1910.03774v3)

Published 9 Oct 2019 in math.OA

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.

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