K-theory of group Banach algebras and Banach property RD

Abstract

We investigate Banach algebras of convolution operators on the $L^p$ spaces of a locally compact group, and their K-theory. We show that for a discrete group, the corresponding K-theory groups depend continuously on $p$ in an inductive sense. Via a Banach version of property RD, we show that for a large class of groups, the K-theory groups of the Banach algebras are independent of $p$.

16 pages. A new theorem (Theorem 1.1) is added. Another proof of Theorem 4.4 is provided. A proof of Proposition 2.3 is provided. Comments are very welcome!