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 ℓ₂​-Betti numbers of a group and surjectivity of different Baum–Connes type assembly maps.