Connected Lie Groups and Property Rd

For a locally compact group, the property of rapid decay (property RD) gives a control on the convolutor norm of any compactly supported function in terms of its L2-norm and the diameter of its support. We characterize the Lie groups that have property RD. 0. Introduction The property of rapid decay (property RD) emerged from the work of U. Haagerup in [15] and was first studied systematically by P. Jolissaint in [21], mostly in the context of finitely generated groups. Property RD gives a control on the convolutor norm of any compactly supported function in terms of its L2-norm and the diameter of its support. Before Haagerup’s work, C. Herz stated and proved in [17, théorème 1] that connected semisimple real Lie groups with finite center have property RD. (Of course, he did not use this terminology.) The terminology rapid decay comes from the fact that a group has property RD if and only if any rapidly decaying function is an L2-convolutor (see Definition 2.3, Lemma 2.4). Property RD is useful in the theory of C*-algebras. A. Connes and H. Moscovici used it in [7] to prove the Novikov conjecture for word hyperbolic groups, and V. Lafforgue used it in [25] to prove the Baum-Connes conjecture for some groups having property (T). Property RD is also relevant to the study of random walks on nonamenable groups. This is used in Section 7 to relate property RD to N. Varopoulos’s work [37] and developed further in [6]. The main result of this article is a precise algebraic description of those connected (real) Lie groups that have property RD. DUKE MATHEMATICAL JOURNAL Vol. 137, No. 3, c © 2007 Received 17 September 2004. Revision received 30 August 2006. 2000 Mathematics Subject Classification. Primary 22D15, 22E30, 43A15; Secondary 46L05. Chatterji’s work partially supported by Swiss Science Foundation grant PA002-101406 and National Science Foundation grant DMS-0405032. Pittet’s work partially supported by Délégation Centre National de la Recherche Scientifique, Université de Provence. Saloff-Coste’s work partially supported by National Science Foundation grant DMS-0102126.