Strong Morita equivalence for completely positive linear maps on $C^*$-algebras

TL;DR

This paper introduces the concept of strong Morita equivalence for completely positive linear maps on $C^*$-algebras, exploring its properties and relationships with bimodule maps, and establishing a correspondence between equivalence classes under Morita equivalence.

Contribution

It defines strong Morita equivalence for completely positive maps, analyzes its properties, and links it to Morita equivalence of $C^*$-algebras, providing a new framework for their classification.

Findings

Strong Morita equivalence for completely positive maps is well-defined and studied.

There is a correspondence between equivalence classes on Morita equivalent $C^*$-algebras.

Equivalent $C^*$-algebras have strongly Morita equivalent classes of completely positive maps.

Abstract

We will introduce the notion of strong Morita equivalence for completely positive linear maps and study its basic properties. Also, we will discuss the relation between strong Morita equivalence for bounded $C^*$-bimodule linear maps and strong Morita equivalence for completely positive linear maps. Furthermore, we will show that if two unital $C^*$-algebras are strongly Morita equivalent, then there is a $1-1$ correspondence between the two sets of all strong Morita equivalence classes of completely positive linear maps on the two unital $C^*$-algebras and we will show that the corresponding two classes of the completely positive linear maps are also strongly Morita equivalent.