  We compute $L^2$-Betti numbers of postliminal, locally compact, unimodular
groups in terms of ordinary dimensions of reduced cohomology with coefficients
in irreducible unitary representations and the Plancherel measure. This allows
us to compute the $L^2$-Betti numbers for semi-simple Lie groups with finite
center, simple algebraic groups over local fields, and automorphism groups of
locally finite trees acting transitively on the boundary.
