Relations between asymptotic and Fredholm representations
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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 $Bπ$ which can be obtained from an asymptotic representation of a discrete group $π$ can be obtained also from a representation of the group $π\times Z$ 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.
LaTeX v2.09, 10 pages, no figures
LaTeX v2.09, 10 pages, no figures