The Minimal Exact Crossed Product

Abstract

Given a locally compact group $G$, we study the smallest exact crossed-product functor $(A,G,\alpha )\mapsto A{⋊}_{\mathcal{E}}G$ on the category of $G$-$C^{\ast }$-dynamical systems. As an outcome, we show that the smallest exact crossed-product functor is automatically Morita compatible, and hence coincides with the functor ${⋊}_{\mathcal{E}}$ 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 $C_{\mathcal{E}}^{\ast }(G)$ 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.

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