Banach algebras associated with spherical representations of the free group

Abstract

We prove that any spherical representation of the free group F weakly contains the regular representation. Moreover C$ , the C*- algebra associated with the spherical representation π, is a compact extension of the reduced C* -algebra of F. We also show that the standard projection onto radial functions admits extensions to C* for a class of representatio ns π of F which includes spherical rep- resentations, as well as the regular representation and the universal representation. Introduction. Let Fr be a free group on r generators x\, ... , xr. Let μ be the finitely supported probability measure equidistributed on {xfι, xfι, ... , xf1}. The operator of convolution by μ is the analogue of the Laplace-Beltrami operator on Riemann rank one sym- metric spaces. By (11) the ^-spectrum of μ can be identified with the ellipse E = {z = x + iy: x2 + (j^y)2 < 1). Any point z of E corresponds in one-to-one fashion to a spherical function φz the eigenfunction of μγ with eigenvalue z. We refer to (11), (7) for this subject. For real z spherical functions are positive definite and give rise to unitary representations of ¥r. Basing our argument on a partic- ular realisation of these representations and on the simplicity of the reduced C*-algebra of F r (10), we prove that all spherical represen- tations weakly contain the regular representation. Moreover the C*- algebras associated with spherical representations are compact exten- sions of C*ed(Fr), the C*-algebra associated with the regular repre- sentation. Finally we consider the standard projection onto radial functions on F r and we prove that it is bounded on any C* -algebra associated with spherical functions.