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

par Bursztyn, Henrique ;Waldmann, Stefan
Référence Journal of geometry and physics, 37, 4, page (307-364)
Publication Publié, 2001-03

Article révisé par les pairs

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Résumé :

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. © 2001 Elsevier Science B.V.