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In this first part of the paper, we define a natural dual object for manifolds with corners and show how pseudodifferential calculus on such manifolds can be constructed in terms of the localization principle in C*-algebras. In the second part, these results will be applied to the solution of Gelfand’s problem on the homotopy classification of elliptic operators for the case of manifolds with corners.

Elliptic Theory on Manifolds with Corners: I. Dual Manifolds and Pseudodifferential Operators