K-theory of reduced C*-algebras and rapidly decreasing functions on groups

We associate to any length function L on a group a space of rapidly decreasing functions on (in the l² sense), denoted by H_L(). When H_L() is contained in the reduced C*-algebra C_r^*() of (), then it is a dense *-subalgebra of C_r^*() and we prove a theorem of A. Connes which asserts that under this hypothesis H_L() has the same K-theory as C_r^*(). We introduce another space of rapidly decreasing functions on (in the l¹ sense), denoted by H_L^1,(), which is always a dense *-subalgebra of the Banach algebra l¹(), and we show that H_L^1,() has the same K-theory as l¹().