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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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