The minimal exact crossed product (Q1717077)

The paper under review builds on previous work by Baum, Guentner and Willett on exotic crossed products for locally compact groups \(G\) [\textit{P. Baum} et al., Ann. \(K\)-Theory 1, No. 2, 155--208 (2016; Zbl 1331.46064)]. The motivation for such constructs is to circumvent the failure of the Baum-Connes conjecture for groups with coefficients by substituting the reduced crossed product functor with the smallest exact crossed product functor that is compatible with Morita equivalences. In the aforementioned paper, such a functor has been shown to exist and, moreover, it recovers all positive past results, while the known counterexamples become confirming examples. The current paper is a continuation of this study and previous work of the authors [J. Reine Angew. Math. 740, 111--159 (2018; Zbl 1400.19003)], answering important questions in this direction. For any crossed product functor \(\rtimes_\mu G\), the authors introduce the \(\mathcal{E}(\mu)\)-completion by the supremum norm arising from applying \(\rtimes_\mu G\) to equivariant short exact sequences. This completion provides the smallest half-exact crossed product functor \(\rtimes_{\mathcal{E}(\mu)} G\) that dominates \(\rtimes_\mu G\). By using tools related to the property C, they show that, if \(\rtimes_\mu G\) is Morita compatible and has the ideal property, then so does \(\rtimes_{\mathcal{E}(\mu)} G\). This has striking consequences when applied to the reduced crossed product. In particular, it follows that \(\rtimes_{\mathcal{E}(r)} G\) is the smallest exact Morita compatible crossed product and thus coincides with the functor \(\rtimes_{\mathcal{E}} G\) of Baum et al. [loc. cit.]. For trivial coefficients, the authors then answer in the positive the question whether \(\mathbb{C} \rtimes_{\mathcal{E}} G = \mathbb{C} \rtimes_{\mathcal{E}(r)} G\) coincides with the reduced group \(C^*\)-algebra. Moving further, they provide a concrete description of \(\rtimes_{\mathcal{E}(r)} G\) for \(C^*\)-algebras that have the weak equivariant lifting property. Finally, the authors show that the smallest exact functor passes by restriction to open subgroups. The latter is in analogy to the case of closed normal subgroups obtained in [\textit{A. Buss} et al., Abel Symp. 12, 61--108 (2016; Zbl 1375.46048)]. The paper closes with a list of questions concerning natural properties of the \(\rtimes_{\mathcal{E}} G\) functor. The paper is well written and self-contained.