Smooth dense subalgebras of reduced group C∗-algebras, Schwartz cohomology of groups, and cyclic cohomology

Abstract

We introduce the notion of Schwartz (co)homology of a discrete group with a length function which is the natural generalization of the bounded cohomology of a discrete group. We prove that if the length function is chosen to be a word length on a group of polynomial growth, then its Schwartz cohomology is canonically isomorphic to the usual group cohomology with complex coefficients. Using this result we will show that the group cohomology of a discrete group of polynomial growth is of polynomial growth in the sense of Connes and Moscovici, and that there are no nontrivial idempotents in the group C ∗-algebra of a torsion free group of polynomial growth by the method of cyclic cohomology of Connes.