Some Groups Whose Reduced C {algebras Have Stable Rank One

It is proved that, for the following classes of groups, ?, the reduced group C-algebra C (?) has stable rank 1: (i) hyperbolic groups which are either torsion{free and non{elementary or which are cocom-pact lattices in a real, noncompact, simple, connected Lie group of real rank 1 having trivial center; (ii) amalgamated free products of groups, ? = G 1 H G 2 , where H is nite and there is g 0 2 ? such that g ?1 0 Hg 0 \ H = f1g. The proofs involve some analysis of the free semigroup property, which is one way of saying that a group ? has an abundance of free sub-semigroups, and of thè 2-spectral radius property, which says that spectral radius of appropriate elements in C (?) may be computed with the 2-norm. x1. Introduction and statement of main results. Let A be a C-algebra with unit. We denote by sr(A) 2 f1; 2; : : :; 1g the stable rank of A and by RR(A) 2 f0; 1; : : :; 1g the real rank of A; deened respectively in Ri83] and BP91]. We recall that one has (i) RR(A) 2sr(A) ? 1 (ii) sr(A) = 1 () GL(A) is dense in A (iii) RR(A) = 0 () GL(A s.a.) is dense in A s.a. where GL(A) respectively A s.a. ; GL(A s.a.)] denotes the group of invertible elements resp. the real subspace of self-adjoint elements, the subset GL(A) \ A s.a. ] in A: Let us also recall that sr(C(()) = 1 2 dim(() + 1 and RR(C(()) = dim(() for the abelian C-algebra of continuous functions on a compact space of dimension dim((); thus, in general, sr(A) and RR(A) may be thought of as \dimensions" of A (or of some dual of A). Several properties follow from the stable rank of A being 1: On the one hand, there are nonstable K{theory properties Ri83], Ri87], Br95] : if p and q are projections in A such that The authors acknowledge support of the Fonds National Suisse de la Recherche Scientiique.