Strong Morita equivalence for conditional expectations

TL;DR

This paper explores the strong Morita equivalence of inclusions of $C^*$-algebras, establishing isometric isomorphisms between their bimodule linear maps and defining related Picard groups.

Contribution

It demonstrates the existence of an isometric isomorphism between spaces of bounded bimodule linear maps under strong Morita equivalence and introduces the Picard group for such maps.

Findings

Existence of isometric isomorphism between bimodule linear map spaces

Properties of the isometric isomorphism analyzed

Definition and discussion of the Picard group for bimodule maps

Abstract

We consider two inclusions of $C^*$-algebras whose small $C^*$-algebras have approximate units of the large $C^*$-algebras and their two spaces of all bounded bimodule linear maps. We suppose that the two inclusions of $C^*$-algebras are strongly Morita equivalent. In this paper, we shall show that there exists an isometric isomorphism from one of the spaces of all bounded bimodule linear maps to the other space and we shall study on basic properties about the isometric isomorphism. And, using this isometric isomorphism, we define the Picard group for a bimodule linear map and discuss on the Picard group for a bimodule linear map.