Journal of Operator Theory

Volume 65, Issue 1, Winter 2011 pp. 145-155.

Convex polytopes and the index of Wiener-Hopf operators

Authors: Alexander Alldridge
Author institution: Institut fuer Mathematik, Universitaet Paderborn, 33098 Paderborn, Germany and Mathematisches Institut, Universitaet zu Koeln, Weyertal 86--90, 50939 Koeln, Germany

Summary: 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 $A_\Omega/\knums$ and $S$ are $KK$-equivalent. Moreover, the isomorphism class of $A_\Omega$ is a complete invariant for the combinatorial type of $P $.

Keywords: Wiener--Hopf operator, $C^*$-algebra, index formula, $K$-theory, $KK$-\break theory, $KK$-contractible, Atiyah--Hirzebruch spectral sequence, polyhedron, convex cone, polyhedral cone, $f$-vector, combinatorial equivalence

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