  Let M be a closed enlargeable spin manifold. We show non-triviality of the
universal index obstruction in the K-theory of the maximal $C^*$-algebra of the
fundamental group of M. Our proof is independent from the injectivity of the
Baum-Connes assembly map for the fundamental group of M and relies on the
construction of a certain infinite dimensional flat vector bundle out of a
sequence of finite dimensional vector bundles on M whose curvatures tend to
zero.
  Besides the well known fact that M does not carry a metric with positive
scalar curvature, our results imply that the classifying map $M \to B \pi_1(M)$
sends the fundamental class of M to a nontrivial homology class in $H_n(B
\pi_1(M) ; \Q)$. This answers a question of Burghelea (1983).
