On -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 { M n } n = k ∞ , 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, is isomorphic to our construction when M = E . If M has an oriented Spin q -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.