Relations between asymptotic and Fredholm representations
Abstract
We prove that for matrix algebras
M
n
there exists a monomorphism
(
∏
n
M
n
/
⊕
n
M
n
)⊗C(
S
1
)→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×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.