Completely positive inner products and strong Morita equivalence

Henrique Bursztyn and Stefan Waldmann

Vol. 222 (2005), No. 2, 201–236

DOI: 10.2140/pjm.2005.222.201

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.

Keywords

complete positivity, strong Morita equivalence, Rieffel induction

Mathematical Subject Classification 2000

Primary: 16D90, 46L08, 53D55

Milestones

Received: 6 April 2004

Accepted: 24 April 2004

Published: 1 December 2005