### On C*-algebras and K-theory for infinite-dimensional Fredholm manifolds, by Jody Trout (Dartmouth College) and Dorin Dumitrascu (University of Arizona)

[Oct 5, 2005: This paper has been accepted for publication in the
journal "Topology and Its Applications". A revised version of the
preprint, to replace the Nov 22, 2004, has been provided.]

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.

Jody Trout (Dartmouth College) <jody.trout@dartmouth.edu>

Dorin Dumitrascu (University of Arizona) <dumitras@gauss.dartmouth.edu>