An index theorem for Wiener--Hopf operators

الأصل · EN

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.