A Thom isomorphism for infinite rank Euclidean bundles

Abstract

An equivariant Thom isomorphism theorem in operator Ktheory is formulated and proven for infinite rank Euclidean vector bundles over finite dimensional Riemannian manifolds. The main ingredient in the argument is the construction of a non-commutative C ? -algebra associated to a bundle E ! M, equipped with a compatible connection r, which plays the role of the algebra of functions on the infinite dimensional total space E. If the base M is a point, we obtain the Bott periodicity isomorphism theorem of Higson-Kasparov-Trout [19] for infinite dimensional Euclidean spaces. The construction applied to an even finite rank spin c -bundle over an even-dimensional proper spin c -manifold reduces to the classical Thom isomorphism in topological K-theory. The techniques involve noncommutative geometric functional analysis.