Asymptotic and Fredholm representations of discrete groups

A $C^*$-algebra servicing the theory of asymptotic representations and its embedding into the Calkin algebra that induces an isomorphism of $K_1$-groups is constructed. As a consequence, it is shown that all vector bundles over the classifying space $B\pi$ that can be obtained by means of asymptotic representations of a discrete group $\pi$ can also be obtained by means of representations of the group $\pi \times {\mathbb Z}$ into the Calkin algebra. A generalization of the concept of Fredholm representation is also suggested, and it is shown that an asymptotic representation can be regarded as an asymptotic Fredholm representation.