The strong Morita equivalence for inclusions of $C^*$-algebras and conditional expectations for equivalence bimodules

TL;DR

This paper introduces and studies the concepts of strong Morita equivalence for unital inclusions of $C^*$-algebras and conditional expectations on equivalence bimodules, exploring their fundamental properties.

Contribution

It defines new notions of strong Morita equivalence and conditional expectations in the context of $C^*$-algebras and bimodules, expanding the theoretical framework.

Findings

Established basic properties of the new notions

Extended Morita theory to include unital inclusions

Analyzed conditional expectations in bimodule contexts

Abstract

We shall introduce the notions of the strong Morita equivalence for unital inclusions of unital $C^*$-algebras and conditional expectations from an equivalence bimodule onto its closed subspace with respect to conditional expectations from unital $C^*$-algebras onto their unital $C^*$-subalgebras. Also, we shall study their basic properties.