  Almost flat finitely generated projective Hilbert C*-module bundles were
successfully used by Hanke and Schick to prove special cases of the Strong
Novikov Conjecture. Dadarlat later showed that it is possible to calculate the
index of a K-homology class $\eta\in K_*(M)$ twisted with an almost flat bundle
in terms of the image of $\eta$ under Lafforgue's assembly map and the almost
representation associated to the bundle. Mishchenko used flat
infinite-dimensional bundles equipped with a Fredholm operator in order to
prove special cases of the Novikov higher signature conjecture.
  We show how to generalize Dadarlat's theorem to the case of an
infinite-dimensional bundle equipped with a continuous family of Fredholm
operators on the fibers. Along the way, we show that special cases of the
Strong Novikov Conjecture can be proven if there exist sufficiently many almost
flat bundles with Fredholm operator.
  To this end, we introduce the concept of an asymptotically flat Fredholm
bundle and its associated asymptotic Fredholm representation, and prove an
index theorem which relates the index of the asymptotic Fredholm bundle with
the so-called asymptotic index of the associated asymptotic Fredholm
representation.
