Generalized Products and Semiclassical Quantization

The notion of a generalized product, refining that of a (symmetric and smooth) simplicial space is introduced and shown to imply the existence of an algebra of pseudodifferential operators. This encompasses many constructions of such algebras on manifolds with corners. The main examples discussed in detail here are related to the semiclassical (and adiabatic) calculus as used in the approach to a twisted form of the Atiyah-Singer index theorem in work with Is Singer and Mathai Varghese.