Relations between asymptotic and Fredholm representations

Details Relations between asymptotic and Fredholm representations Relations between asymptotic and Fredholm representations

1997-07-16

arxiv.org/abs/funct-an/9707005v1

Mathematics

Operator Algebras

LaTeX v2.09, 10 pages, no figures

Scientific paper

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 $B\pi$ 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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