Higher Kazhdan projections, ${\ell }_2$-Betti numbers and Baum–Connes conjectures

Abstract

We introduce higher-dimensional analogs of Kazhdan projections in matrix algebras over group $C^{\ast }$-algebras and Roe algebras. These projections are constructed in the framework of cohomology with coefficients in unitary representations and in certain cases give rise to non-trivial $K$-theory classes. We apply the higher Kazhdan projections to establish a relation between ${\ell }_2$-Betti numbers of a group and surjectivity of different Baum–Connes type assembly maps.