Algebraic Rieffel Induction, Formal Morita Equivalence, and Applications to Deformation Quantization

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. [email protected] Research supported by a fellowship from CNPq, Grant 200481/96-7. [email protected] Research supported by the Communauté française de Belgique, through an Action de Recherche Concertée de la Direction de la Recherche Scientifique.