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 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
non-commutative geometric functional analysis.
This paper has appeared in
Homology, Homotopy and Applications, vol 5, no. 1, article 7, pp 121-159.
Jody Trout <email@example.com>