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 under stood 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. (C) 2001 Elsevier Science B.V, All rights reserved. MSC: 58B15; 81S99.