Group C*-algebras as decreasing intersection of nuclear C*-algebras

Abstract

We prove that for every exact discrete group $\Gamma$, there is an intermediate C*-algebra between the reduced group C*-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*-algebra is realized in this way. We also study extensions of the reduced free group C*-algebras and show that any exact absorbing or unital absorbing extension of it by any stable separable nuclear C*-algebra is realized in this way.

22 pages. More detailed proofs. No substantial change. To appear in Amer. J. Math. Expanded version of arXiv:1406.2740

References in corpus (1)

• Amenable minimal Cantor systems of free groups arising from diagonal actions

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• The approximation property and exactness of locally compact groups
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• Crossed product splitting of intermediate operator algebras via 2-cocycles
• Splitting of Tensor Products and Intermediate Factor Theorem: Continuous Version
• Invariant $C^{\ast }$-subalgebras of the reduced group $C^{\ast }$-algebra