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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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