Abstract

We prove that for every exact discrete group $\Gamma$, there is an intermediate ${\rm C}^\ast$-algebra between the reduced group ${\rm C}^\ast$-algebra and the intersection of the group von Neumann algebra and the uniform Roe algebra which is realized as the intersection of a decreasing sequence of isomorphs of the Cuntz algebra ${\cal O}_2$. In particular, when $\Gamma$ has the AP (approximation property), the reduced group ${\rm C}^\ast$-algebra is realized in this way. We also study extensions of the reduced free group ${\rm C}^\ast$-algebras and show that any exact absorbing or unital absorbing extension of it by any stable separable nuclear ${\rm C}^\ast$-algebra is realized in this way.

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