## 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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