  Our main result provides a closed expression for the completely bounded
Fourier multiplier norm of the spherical functions on the generalized Lorentz
groups. As a corollary, we find that there is no uniform bound on the
completely bounded Fourier multiplier norm of the spherical functions on the
generalized Lorentz groups. We extend the latter result to the remaining
connected, real rank one, simple Lie groups with finite center, and as an
application we obtain that each of the above mentioned groups has a completely
bounded Fourier multiplier, which is not the coefficient of a uniformly bounded
representation of the group on a Hilbert space.
