Algebraic Rieffel induction, formal Morita equivalence, and applications to deformation quantization
✍ Scribed by Henrique Bursztyn; Stefan Waldmann
- Publisher
- Elsevier Science
- Year
- 2001
- Tongue
- English
- Weight
- 385 KB
- Volume
- 37
- Category
- Article
- ISSN
- 0393-0440
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✦ Synopsis
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.