Permanence properties of \(C^*\)-exact groups.

*(English)*Zbl 0958.46036For a locally compact group \(G,\) let \({\mathcal C}^*_G\) be the category whose objects are the pairs \((A,\alpha)\) consisting of a \(C^*\)-algebra \(A\) with a continuous \(G\)-action \(\alpha,\) and whose morphisms are the \(G\)-equivariant \(*\)-homomorphisms. For \((A,\alpha)\) in \(C^*_G,\) let \(A\times_{\alpha,r} G\) denote the reduced crossed product. The authors introduced in [Math. Ann. 315, No. 2, 169-203 (1999; Zbl 0946.46054)] the following definition: \(G\) is called exact if the functor \(A\to A\times_{\alpha,r} G\) for \((A,\alpha)\) in \({\mathcal C}^*_G\) is short-exact (that is, if it preserves short exact sequences in the category \({\mathcal C}^*_G\)). They showed in the above mentioned article, among other things, that a discrete group \(G\) is exact if and only if its reduced \(C^*\)-algebra is exact. (A \(C^*\)-algebra \(B\) is exact if the functor \(A\to A\otimes B,\) defined on the category of all \(C^*\)-algebras, is short-exact, where \(\otimes\) denotes the spatial tensor product. This notion is due to E. Kirchberg [see for instance J. Oper. Theory 10, 3-8 (1983; Zbl 0543.46035)].)

In the article under review, the authors show that the class of exact groups is closed under various operations such as passing to a closed subgroup and taking extensions. Using these results, they prove that almost-connected groups are exact.

Reviewer’s remark: The authors ask whether all locally compact groups are exact. In the meantime, progress has been made on this question. M. Gromov announced the existence of certain “exotic” discrete groups which, by work of N. Ozawa [C. R. Acad. Sci., Paris, Sér I 330, 691-695 (2000)], are not exact.

In the article under review, the authors show that the class of exact groups is closed under various operations such as passing to a closed subgroup and taking extensions. Using these results, they prove that almost-connected groups are exact.

Reviewer’s remark: The authors ask whether all locally compact groups are exact. In the meantime, progress has been made on this question. M. Gromov announced the existence of certain “exotic” discrete groups which, by work of N. Ozawa [C. R. Acad. Sci., Paris, Sér I 330, 691-695 (2000)], are not exact.

Reviewer: M.B.Bekka (Metz)