On C*-algebras and K-theory for infinite-dimensional Fredholm Manifolds

Abstract

Let M be a smooth Fredholm manifold modeled on a separable infinite-dimensional Euclidean space E with Riemannian metric g. Given an (augmented) Fredholm filtration F of M by finite-dimensional submanifolds (M_n), we associate to the triple (M, g, F) a non-commutative direct limit C*-algebra A(M, g, F) = lim A(M_n) that can play the role of the algebra of functions vanishing at infinity on the non-locally compact space M. The C*-algebra A(E), as constructed by Higson-Kasparov-Trout for their Bott periodicity theorem for infinite dimensional Euclidean spaces, is isomorphic to our construction when M = E. If M has an oriented Spin_q-structure (1 <= q <=\infty), then the K-theory of this C*-algebra is the same (with dimension shift) as the topological K-theory of M defined by Mukherjea. Furthermore, there is a Poincare' duality isomorphism of this K-theory of M with the compactly supported K-homology of M, just as in the finite-dimensional spin setting.