Convex polytopes and the index of Wiener-Hopf operators

Abstract

We study the C$^*$-algebra of Wiener-Hopf operators $A_\Omega$ on a cone $\Omega$ with polyhedral base $P$. As is known, a sequence of symbol maps may be defined, and their kernels give a filtration by ideals of $A_\Omega$, with liminary subquotients. One may define $K$-group valued 'index maps' between the subquotients. These form the $E^1$ term of the Atiyah-Hirzebruch type spectral sequence induced by the filtration. We show that this $E^1$ 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 $A_\Omega$ is $KK$-contractible, and that $A_\Omega/\mathbb K$ and $S$ are $KK$-equivalent. Moreover, the isomorphism class of $A_\Omega$ is a complete invariant for the combinatorial type of $P$.