An index theorem for Wiener–Hopf operators

We study the multivariate generalisation of the classical Wiener–Hopf algebra, which is the C∗-algebra generated by the Wiener–Hopf operators, given by 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. It admits a composition series, and therefore, a ‘symbol’ calculus. Using groupoid methods, we obtain, in the framework of Kasparov's bivariant KK-theory, a topological expression of the index maps associated to these symbol maps in terms of geometric-topological data of the underlying convex cone. This generalises an index theorem by Upmeier concerning Wiener–Hopf operators on symmetric cones. Our result covers a wide class of cones containing polyhedral and homogeneous cones.