00 5 v 1 1 6 Ju l 1 99 7 Relations between asymptotic and Fredholm representations ∗

We prove that for matrix algebras Mn there exists a monomorphism ( ∏ nMn/⊕n Mn)⊗ C(S ) −→ Q into the Calkin algebra which induces an isomorphism of the K1-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×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. 1 Asymptotic representations as representations into the Calkin algebra Let π be a discrete finitely presented group, and let F ⊂ π be a finite subset. Denote by U(n) the unitary group of dimension n and fix a number ε > 0. Definition 1.1 A map σ : π −→ U(n) is called an ε-almost representation with respect to F if σ(g) = σ(g) holds for all g ∈ π and if ‖σ‖F = sup{‖σ(gh)− σ(g)σ(h)‖ : g, h, gh ∈ F} ≤ ε. Let {nk} be a strictly increasing sequence of positive integers and let σ = {σk : π −→ U(nk)} be a sequence of εk-almost representations. We assume that the groups U(nk) are embedded into the groups U(nk+1) in the standard way, so it makes possible to compare almost representations for different k. Then we can consider the maps σk ⊕ 1 : π −→ U(nk)⊕ U(nk+1 − nk) −→ U(nk+1), which we also denote by σk. Definition 1.2 A sequence of εk-almost representations is called an asymptotic representation of the group π (with respect to the finite subset F and a sequence {nk}) if the sequences εk and ‖σk(g)− σk+1(g)‖ : g ∈ F ⊂ π tend to zero. ∗This research was partially supported by RFBR (grant No 96-01-00276) and by DFG.