Some Remarks Concerning the Baum-Connes Conjecture

P. Baum and A. Connes have made a deep conjecture about the calculation of the K-theory of certain types of C∗-algebras [1, 2]. In particular, for a discrete group Γ they have conjectured the calculation of K∗(C r (Γ)), the Ktheory of the reduced C∗-algebra of Γ. So far, there is quite little evidence for this conjecture. For example, there is not a single property T group for which it is known to be true. In this note we show that, in some sense, the homological algebra of their conjecture is correct. In many cases, the periodic cyclic homology of certain dense subalgebras suggests what the K-theory should be. In the case of a discrete group Γ, the periodic cyclic homology of the algebraic group algebra CΓ is quite easy to calculate. Let 〈Γ〉f denote the set of conjugacy classes of elements of finite order, and let 〈Γ〉i denote the set of conjugacy classes of infinite order. For γ ∈ Γ, let Γγ denote the centralizer of γ in Γ. Let Γγ/γ be the quotient of Γγ by the cyclic subgroup generated by γ. If γ is of finite order, then Hi(BΓγ/γ; C) ∼= Hi(BΓ; C). (Note that for a discrete group BΓ = K(Γ, 1); we will freely use both notations.) If γ is of infinite order, then