Star products and local line bundles
  The notion of a local line bundle on a manifold, classified by 2-cohomology with real coefficients, is introduced. The twisting of pseudodifferential operators by such a line bundle leads to an algebroid with elliptic elements with real-valued index, given by a twisted variant of the Atiyah-Singer index formula. Using ideas of Boutet de Monvel and Guillemin the corresponding twisted Toeplitz algebroid on any compact symplectic manifold is shown to yield the star products of Lecomte and DeWilde ([MR84g:17014]) see also Fedosov's construction in [MR92k:58267]. This also shows that the trace on the star algebra is identified with the residue trace of Wodzicki and Guillemin 
Richard Melrose