Algebraic Rieffel induction, formal Morita equivalence, and applications to deformation quantization

Abstract

In this paper we consider algebras with involution over a ring C which is given by the quadratic extension by i of an ordered ring R. We discuss the * -representation theory of such * -algebras on pre-Hilbert spaces over C and develop the notions of Rieffel induction and formal Morita equivalence for this category analogously to the situation for C * -algebras. Throughout this paper the notion of positive functionals and positive algebra elements will be crucial for all constructions. As in the case of C * -algebras, we show that the GNS construction of * -representations can be understood as Rieffel induction and, moreover, that formal Morita equivalence of two * -algebras, which is defined by the existence of a bimodule with certain additional structures, implies the equivalence of the categories of strongly non-degenerate * -representations of the two * -algebras. We discuss various examples like finite rank operators on pre-Hilbert spaces and matrix algebras over * -algebras. Formal Morita equivalence is shown to imply Morita equivalence in the ring-theoretic framework. Finally we apply our considerations to deformation theory and in particular to deformation quantization and discuss the classical limit and the deformation of equivalence bimodules. *

Completely positive inner products and strong Morita equivalence

“…* -Algebras possessing a "large" amount of positive linear functionals, such as C * -algebras and formal hermitian deformation quantizations [Bursztyn and Waldmann 2001a;2001b], are such that (3-4) and (3-5) coincide.…”

“…We now consider * -representations of * -algebras on inner-product modules, extending the discussion in [Bursztyn and Waldmann 2001a;Bursztyn and Waldmann 2001b;Bordemann and Waldmann 1998]. Let Ꮽ be a * -algebra over C, and let Ᏼ be an inner-product Ᏸ-module.…”

“…Their connection is suggested by their common role as "quantum algebras" in mathematical physics, despite the fact that the former is a purely algebraic notion, whereas the latter has important analytical features. Building on [Bursztyn and Waldmann 2001a;2001b], we develop in this paper a framework for their unified study, focusing on Morita theory; in particular, the properties shared by C * -algebras and star products allow us to develop a general theory of strong Morita equivalence in which they are treated in equal footing.…”

“…see [Bursztyn and Waldmann 2001a]. We use the convention that · , · is linear in the second argument.…”

“…These are direct analogues of complex pre-Hilbert spaces. A * -representation of a * -algebra Ꮽ on a pre-Hilbert space Ᏼ is a * -homomorphism from Ꮽ into the adjointable endomorphisms B(Ᏼ) of Ᏼ [Bursztyn and Waldmann 2001a;2001b]; the main examples are the usual representations of C * -algebras on Hilbert spaces and the formal representations of star products. See, for instance, [Bordemann and Waldmann 1998;Waldmann 2002].…”