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Expanders, exact crossed products, and the Baum-Connes conjecture. (English) Zbl 1331.46064
For a second countable locally compact group $$G$$, the Baum-Connes conjecture with coefficients (BCC) asserts that the Baum-Connes assembly map $$K_*^{\text{top}}(G;A)\to K_*(A\rtimes_{\text{red}}G)$$ is an isomorphism for any $$C^*$$-algebra $$A$$ equipped with a continuous action of $$G$$. Counterexamples to this conjecture related to failure of exactness were found by Higson, Lafforgue and Skandalis in [N. Higson et al., Geom. Funct. Anal. 12, No. 2, 330–354 (2002; Zbl 1014.46043)]. To avoid these counterexamples, the authors reformulate the BCC, replacing the reduced crossed product by a minimal exact and the Morita-compatible crossed product $$\rtimes_{\mathcal E}$$ (which is proved to exist). Their reformulation asserts that the corresponding assembly map $$K_*^{\text{top}}(G;A)\to K_*(A\rtimes_{\mathcal E}G)$$ is an isomorphism. The reformulated conjecture holds true for all groups satisfying the original BCC, and also for some expander-based counterexamples to BCC.
The authors also discuss the reformulated BCC for groupoids and relations of their crossed product to some other examples of exotic crossed products.

##### MSC:
 46L80 $$K$$-theory and operator algebras (including cyclic theory) 22D25 $$C^*$$-algebras and $$W^*$$-algebras in relation to group representations 46L85 Noncommutative topology 58B34 Noncommutative geometry (à la Connes)
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