A Thom isomorphism for infinite rank Euclidean bundles

Abstract

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 $\mathfrak{E} \to M$, equipped with a compatible connection $\nabla$, which plays the role of the algebra of functions on the infinite dimensional total space $\mathfrak{E}$. 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 $\rm{spin}^c$-bundle over an even-dimensional proper $\rm{spin}^c$-manifold reduces to the classical Thom isomorphism in topological $K$-theory. The techniques involve non-commutative geometric functional analysis.