Relations between asymptotic and Fredholm representations

Abstract

We prove that for matrix algebras $M_n$ there exists a monomorphism $({\prod }_n{M}_n/{\oplus }_n{M}_n)\otimes C(S^1)\to \mathcal{Q}$ into the Calkin algebra which induces an isomorphism of the $K_1$-groups. As a consequence we show that every vector bundle over a classifying space $B\pi$ which can be obtained from an asymptotic representation of a discrete group $\pi$ can be obtained also from a representation of the group $\pi \times Z$ into the Calkin algebra. We give also a generalization of the notion of Fredholm representation and show that asymptotic representations can be viewed as asymptotic Fredholm representations.

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