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Algebraic Rieffel induction, formal Morita equivalence, and applications to deformation quantization. (English) Zbl 1039.46052
Let $$R$$ be an ordered ring and $$C$$ its quadratic extension $$C=R+iR$$, where $$i^ 2=-1$$. The main goal of the extensive work under review is to develop a purely algebraic theory of $$\ast$$-algebras over $$C$$ analogous to theory of $$C^\ast$$-algebras. At first the positive elements of a $$\ast$$-algebra $$A$$ over $$C$$ and positive functionals on $$A$$ are studied. The analog of the GNS construction and a purely algebraic version of Rieffel induction is obtained. Formal Morita equivalence of two $$\ast$$-algebras defined by the existence of a bimodule with certain additional structure is investigated. Various examples of finite rank operators on pre-Hilbert spaces and matrix algebras over $$\ast$$-algebras are studied. Finally, the results of the paper are applied to deformation theory and in particular to deformation quantization.

MSC:
 46L55 Noncommutative dynamical systems 53D55 Deformation quantization, star products 16D90 Module categories in associative algebras 22D30 Induced representations for locally compact groups
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References:
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