Geodesic
The Minimal Exact Crossed Product
Documenta mathematica, Tome 23 (2018), pp. 2043-2077.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Given a locally compact group $G$, we study the smallest exact crossed-product functor $(A,G,\alpha)\mapsto A\rtimes_{\mathcal E} G$ on the category of $G$-$C^*$-dynamical systems. As an outcome, we show that the smallest exact crossed-product functor is automatically Morita compatible, and hence coincides with the functor $\rtimes_{\mathcal E}$ as introduced by P. Baum et al. in their reformulation of the Baum-Connes conjecture [Ann. $K$-Theory 1, No. 2, 155--208 (2016; Zbl 1331.46064)]. We show that the corresponding group algebra $C_{\mathcal E}^*(G)$ always coincides with the reduced group algebra, thus showing that the new formulation of the Baum-Connes conjecture coincides with the classical one in the case of trivial coefficients.
Classification : 46L55, 46L08, 46L80
Keywords: exotic crossed products, Baum-Connes conjecture, exact groups, exotic group algebras 
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     title = {The {Minimal} {Exact} {Crossed} {Product}},
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Buss, Alcides; Echterhoff, Siegfried; Willett, Rufus. The Minimal Exact Crossed Product. Documenta mathematica, Tome 23 (2018), pp. 2043-2077. http://geodesic.mathdoc.fr/item/DOCMA_2018__23__a4/