Index Theory for Wiener–Hopf Operators on Convex Cones

What is it about?

We consider Wiener–Hopf operators with continuous symbol, defined on the L2 space of a convex cone (w.r.t. Lebesgue measure), and the C*-algebra of bounded operators generated by them. In the (classical) case of a Wiener–Hopf operator on the half line, the property of being Fredholm can be characterised in terms of the symbol, and if Fremdholmness obtains, the numerical index can be computed in terms of topological data and the symbol. We show how to extend this theory to a large class of cones characterised by two topologial conditions. For the special case of polyhedral cones, we find an interesting connection between the homological cellular differential and the index maps associated with the corresponding C*-algebra. This work was partly jointly conducted with Troels Johansen of the University of Kiel.

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