Relations between asymptotic and Fredholm representations

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 {cal 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 $Bpi$ which can be obtained from an asymptotic representation of a discrete group $pi$ can be obtained also from a representation of the group $pitimes Z$ 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.