Convex Polytopes and the Index of Wiener–hopf Operators

We study the C∗-algebra of Wiener–Hopf operators AΩ on a cone Ω with polyhedral base P. As is known, a sequence of symbol maps may be defined, and their kernels give a filtration by ideals of AΩ, with liminary subquotients. One may define K-group valued “index maps” between the subquotients. These form the E1 term of the Atiyah–Hirzebruch type spectral sequence induced by the filtration. We show that this E1 term may, as a complex, be identified with the cellular complex of P, considered as CW-complex by taking convex faces as cells. It follows that AΩ is KK-contractible, and that AΩ/K and S are KK-equivalent. Moreover, the isomorphism class of AΩ is a complete invariant for the combinatorial type of P.