An index theorem for Wiener--Hopf operators

Details An index theorem for Wiener--Hopf operators An index theorem for Wiener--Hopf operators

2006-11-07

arxiv.org/abs/math/0611198v3

Adv. Math. 218 (2008), no. 1, 163--201

Mathematics

Operator Algebras

46 pages, 1 figure; last version prior to publication, journal reference added

Scientific paper

10.1016/j.aim.2007.11.024

We study multivariate generalisations of the classical Wiener--Hopf algebra, which is the C$^*$-algebra generated by the Wiener--Hopf operators, given by the convolutions restricted to convex cones. By the work of Muhly and Renault, this C$^*$-algebra is known to be isomorphic to the reduced C$^*$-algebra of a certain restricted action groupoid. In a previous paper, we have determined a composition series of this C$^*$-algebra, and compute the $K$-theory homomorphisms induced by the `symbol' maps given by the subquotients of the composition series in terms of the analytical index of a continuous family of Fredholm operators. In this paper, we obtain a topological expression for these index maps in terms of geometric-topological data naturally associated to the underlying convex cone. The resulting index formula is expressed in the framework of Kasparov's bivariant $KK$-theory. Our proof relies heavily on groupoid methods.

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