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

$P$ .

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