The minimal exact crossed product

TL;DR

This paper introduces and analyzes the smallest exact crossed-product functor for group actions on C*-algebras, showing it aligns with known formulations of the Baum-Connes conjecture and coincides with the reduced group algebra.

Contribution

It establishes the properties of the minimal exact crossed-product functor, including Morita compatibility and its equivalence with the functor used in Baum, Guentner, and Willett's reformulation of the Baum-Connes conjecture.

Findings

The minimal exact crossed-product functor is Morita compatible.

The associated group algebra coincides with the reduced group algebra.

The reformulation of the Baum-Connes conjecture matches the classical version for trivial coefficients.

Abstract

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 Baum, Guentner, and Willett in their reformulation of the Baum-Connes conjecture (see [2]). 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. Erratum: After publication of this manuscript, some gaps have unfortunately been found affecting some parts of the paper. We therefore included an appendix with an erratum at the end of this paper explaining the mistakes and keeping the original published version unchanged.