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Smooth dense subalgebras of reduced group $$C^*$$-algebras, Schwartz cohomology of groups, and cyclic cohomology. (English) Zbl 0787.46043
Summary: 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.

##### MSC:
 46L05 General theory of $$C^*$$-algebras 46M20 Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.) 46L85 Noncommutative topology
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