On C-algebras and K-theory for Infinite-dimensional Fredholm Manifolds

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 {Mn}∞n=k, we associate to the triple (M, g,F) a non-commutative direct limit C-algebra A(M, g,F) = lim −→ A(Mn) 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, is isomorphic to our construction when M = E. If M has an oriented Spinq-structure (1 ≤ q ≤ ∞), 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 Poincaré duality isomorphism of this K-theory of M with the compactly supported K-homology of M , just as in the finite-dimensional spin setting.