On C∗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 EE with Riemannian metric g  . Given an augmented Fredholm filtration FF of M   by finite-dimensional submanifolds {Mn}n=k∞, we associate to the triple (M,g,F)(M,g,F) a non-commutative direct limit C∗C∗-algebraA(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∗C∗-algebra A(E)A(E), as constructed by Higson–Kasparov–Trout for their Bott periodicity theorem, is isomorphic to our construction when M=EM=E. If M   has an oriented SpinqSpinq-structure (1⩽q⩽∞)(1⩽q⩽∞), then the K  -theory of this C∗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.