Completely positive inner products and strong Morita equivalence

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.

Induction of Representations in Deformation Quantization

“…In order to give a unified treatment of C * -algebras and the * -algebras over C [[λ]] defined by Hermitian star products, we work in the following general algebraic setting, see [10] for details: we consider * -algebras A over a ring of the form C = R(i); here R is an ordered ring, like e.g. R or R [[λ]], so C is a ring extension of R by a square root of −1.…”

“…[13]. The reader may consult [10] for details. Let A be a * -algebra over C, and let E be a (right) A-module (we may write E A to stress the A-action).…”

“…An A-valued inner product on E satisfying the condition in (2) is called completely positive [10]. If ·, · A is completely positive, we call (E, ·, · A ) a pre-Hilbert A-module; a (B, A)-inner product bimodule for which the Avalued inner product is completely positive is called a pre-Hilbert bimodule.…”

“…Thus (E 0 , h 0 ) comes from a vector bundle E over M carrying a Hermitian fibre metric h 0 , and (E, h) is an example of a deformation quantization of a Hermitian vector bundle in the sense of [5]. By [10], it follows that (E, h) is isometric to an A-module as the one in Example 3.3, part (3). Hence h is completely positive.…”

Hermitian Star Products are Completely Positive Deformations

“…In particular, Theorem 2.1 has direct applications to the theory of strong Morita equivalence of star products, see [8].…”