Strong Morita equivalences for completely positive linear maps and GNS-C*-correspondences

TL;DR

This paper introduces a notion of strong Morita equivalence for completely positive linear maps on unital C*-algebras and explores their relationship with GNS-C*-correspondences, establishing a correspondence between equivalence classes.

Contribution

It defines strong Morita equivalence for these maps and links it to the equivalence of GNS-C*-correspondences, providing a new framework for their classification.

Findings

Establishes a bijective correspondence between strong Morita equivalence classes of maps for Morita equivalent algebras.

Connects strong Morita equivalence of maps with that of GNS-C*-correspondences.

Provides a structural understanding of completely positive maps via Morita theory.

Abstract

We will consider the set of all completely positive linear maps from a unital $C^*$-algebra to the $C^*$-algebra of all (bounded) adjointable right Hilbert $C^*$-module maps, which are automatically bounded, on a right Hilbert $C^*$-module and we will introduce strong Morita equivalence for elements in this set. In this paper, we will give the following result: If two classes of two unital $C^*$-algebras are strongly Morita equivalent, respectively, then we can construct a bijective correspondence between two sets of all strong Morita equivalence classes of completely positive linear maps given as above. Furthermore, we will discuss the relation between strong Morita equivalence for completely positive linear maps and strong Morita equivalence for GNS-$C^*$-correspondences.