Group C*-algebras without the completely bounded approximation property

Abstract

It is proved that: (1) The Fourier algebra A(G) of a simple Lie group G of real rank at least 2 with finite center does not have a multiplier bounded approximate unit. (2) The reduced C*-algebra of any lattice in a non-compact simple Lie group of real rank at least 2 with finite center does not have the completely bounded approximation property. Hence, the results obtained by J. de Canniere and the author for SO(n,1), n at least 2, and by M. Cowling for SU(n,1) do not generalize to simple Lie groups of real rank at least 2.