Deformation quantization of Hermitian vector bundles

Motivated by deformation quantization, we consider in this paper *-algebras cal A over rings sf C=R(i), where sf 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) cal A-valued inner product. For A=C-infinity(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-infinity(M) and Gamma (infinity)(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.