Deformation Quantization of Hermitian Vector Bundles

Motivated by deformation quantization, we consider in this paper -algebras A over rings C = R(i), where R is an ordered ring and i = −1, and study the deformation theory of projective modules over these algebras carrying the additional structure of a (positive) A-valued inner product. For A = C(M), M a manifold, these modules can be identified with Hermitian vector bundles E overM . We show that for a fixed Hermitian star-product onM , 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(M) and Γ(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. [email protected] Research supported by a fellowship from CNPq, Grant 200481/96-7. [email protected] Research supported by the Communauté française de Belgique, through an Action de Recherche Concertée de la Direction de la Recherche Scientifique.