Group \(C^\ast\)-algebras as decreasing intersection of nuclear \(C^\ast\)-algebras. (English) Zbl 1390.46052
Summary: We prove that for every exact discrete group \(\Gamma\), there is an intermediate \(C^\ast\)-algebra between the reduced group \(C^\ast\)-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 \({\mathcal O}_2\). In particular, when \(\Gamma\) has the AP (approximation property), the reduced group \(C^\ast\)-algebra is realized in this way. We also study extensions of the reduced free group \(C^\ast\)-algebras and show that any exact absorbing or unital absorbing extension of it by any stable separable nuclear \(C^\ast\)-algebra is realized in this way.