Almost flat K-theory of classifying spaces

Abstract

We give a rigorous account and prove continuity properties for the correspondence between almost flat bundles on a triangularizable compact connected space and the quasi-representations of its fundamental group. For a discrete countable group $\Gamma$ with finite classifying space $B\Gamma$, we study a correspondence between almost flat K-theory classes on $B\Gamma$ and group homomorphism $K_0(C^{\ast }(\Gamma ))\to \mathbb{Z}$ that are implemented by pairs of discrete asymptotic homomorphisms from $C^{\ast }(\Gamma )$ to matrix algebras.