Geodesic
Asymptotic and Fredholm representations of discrete groups
Sbornik. Mathematics, Tome 189 (1998) no. 10, pp. 1485-1504

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A $C^*$-algebra servicing the theory of asymptotic representations and its embedding into the Calkin algebra that induces an isomorphism of $K_1$-groups is constructed. As a consequence, it is shown that all vector bundles over the classifying space $B\pi$ that can be obtained by means of asymptotic representations of a discrete group $\pi$ can also be obtained by means of representations of the group $\pi \times {\mathbb Z}$ into the Calkin algebra. A generalization of the concept of Fredholm representation is also suggested, and it is shown that an asymptotic representation can be regarded as an asymptotic Fredholm representation. 
@article{SM_1998_189_10_a2,
     author = {V. M. Manuilov and A. S. Mishchenko},
     title = {Asymptotic and {Fredholm} representations of discrete groups},
     journal = {Sbornik. Mathematics},
     pages = {1485--1504},
     publisher = {mathdoc},
     volume = {189},
     number = {10},
     year = {1998},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1998_189_10_a2/}
}
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V. M. Manuilov; A. S. Mishchenko. Asymptotic and Fredholm representations of discrete groups. Sbornik. Mathematics, Tome 189 (1998) no. 10, pp. 1485-1504. http://geodesic.mathdoc.fr/item/SM_1998_189_10_a2/