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The notion of a generalized product, refining that of a (symmetric andsmooth) simplicial space is introduced and shown to imply the existence of analgebra of pseudodifferential operators. This encompasses many constructions ofsuch algebras on manifolds with corners. The main examples discussed in detailhere are related to the semiclassical (and adiabatic) calculus as used in theapproach to a twisted form of the Atiyah-Singer index theorem in work with IsSinger and Mathai Varghese.

Generalized Products and Semiclassical Quantization