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

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.