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Deformation quantization of Hermitian vector bundles. (English) Zbl 0982.53073
Summary: Motivated by deformation quantization, we consider in this paper \(*\)-algebras \(\mathcal A\) over rings \(C = R(i)\), where \(R\) is an ordered ring and \(i^2 = -1\), and study the deformation theory of projective modules over these algebras carrying the additional structure of a (positive) \(\mathcal A\)-valued inner product. For \(A = C^\infty(M)\), \(M\) a manifold, these modules can be identified with Hermitian vector bundles \(E\) over \(M\). We show that for a fixed Hermitian star product on \(M\), these modules can always be deformed in a unique way, up to (isometric) equivalence. We observe that there is a natural bijection between the sets of equivalence classes of local Hermitian deformations of \(C^\infty(M)\) and \(\Gamma^\infty(\text{End}(E))\) and that the corresponding deformed algebras are formally Morita equivalent, an algebraic generalization of strong Morita equivalence of \(C^*\)-algebras. We also discuss the semi-classical geometry arising from these deformations.

MSC:
53D55 Deformation quantization, star products
81R60 Noncommutative geometry in quantum theory
53D17 Poisson manifolds; Poisson groupoids and algebroids
81S10 Geometry and quantization, symplectic methods
32L05 Holomorphic bundles and generalizations
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