Completely positive inner products and strong Morita equivalence

TL;DR

This paper develops a unified framework for studying strong Morita equivalence across $C^*$-algebras and hermitian star products, revealing their relation and differences through geometric and cohomological analysis.

Contribution

It introduces a general approach to compare strong and ring-theoretic Morita equivalences, analyzing their Picard groups and geometric implications for star products.

Findings

Strong Morita equivalence induces the same relation for $C^*$-algebras and star products.

Different Picard groups are associated with the two notions of Morita equivalence.

Geometric and cohomological tools describe the differences in star product cases.

Abstract

We develop a general framework for the study of strong Morita equivalence in which $C^*$-algebras and hermitian star products on Poisson manifolds are treated in equal footing. We compare strong and ring-theoretic Morita equivalences in terms of their Picard groupoids for a certain class of unital $^*$-algebras encompassing both examples. Within this class, we show that both notions of Morita equivalence induce the same equivalence relation but generally define different Picard groups. For star products, this difference is expressed geometrically in cohomological terms.