  We introduce higher-dimensional analogs of Kazhdan projections in matrix
algebras over group $C^*$-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.
