The Minimal Exact Crossed Product

Given a locally compact group G , we study the smallest exact crossed-product functor (A,G,α)↦ AE 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 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 CE^*(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.