A Thom Isomorphism for Infinite Rank Euclidean Bundles

An equivariant Thom isomorphism theorem in operator K-theory 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 ∇, 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-bundle over an evendimensional proper spin-manifold reduces to the classical Thom isomorphism in topological K-theory. The techniques involve non-commutative geometric functional analysis.