Subexponential group cohomology and the K - theory of Lafforgue ’ s algebra A max ( π )

Let K∗(X) denote the generalized homology of X with coefficients in the spectrum K (C) [1], π a finitely generated discrete group and Bπ its classifying space. For any Banach algebra A(π) with C[π ] ⊆ A(π), there is an assembly map K∗(Bπ) → K t ∗(A(π)) for the topological K -theory of A(π). In this paper, we are interested in the “maximal unconditional completion” of C[π ] in the reduced group C∗-algebra C∗ r (π). This algebra, denoted Amax(π), was introduced by Lafforgue [10] (the definition is recalled below). The main result of the paper is as follows.