Permanence properties of \(C^*\)-exact groups
For 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 \textit{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 \textit{N. Ozawa} [C. R. Acad. Sci., Paris, Sér I 330, 691-695 (2000)], are not exact.
scientific article; zbMATH DE number 1367404