Completely Bounded Multiplier Norm of Spherical Functions on the Generalized Lorentz Groups

Plot of ‖ϕs‖M0A(G) as a function of s for G = SO0(1, 2). Previously it was known which spherical functions on the general-ized Lorentz groups that were completely bounded multipliers of the corresponding Fourier algebra, but the actual value of the completely bounded multiplier norms were not known. The first part of the talk is a calculation of these norms for all spherical functions on the general-ized Lorentz groups. The second part is a general result on coefficients of uniformly bounded representations of locally compact groups say-ing the following: Given a Gelfand pair (G,K) then the completely bounded spherical functions on (G,K) all have completely bounded multiplier norms less than some fixed constant if there are completely bounded multipliers of G which are not coefficients of strongly con-tinues uniformly bounded representations. These results have been obtained as part of a joint work with Uffe Haagerup. As a corollary to these two parts we conclude that there are com-pletely bounded multipliers of the Fourier algebra of the generalized Lorentz groups, which are not coefficients of strongly continues uni-formly bounded representations. The corresponding result for non-commutative free groups has been proved in 1985 by Haagerup (un-published notes) and also follows as a corollary to a more general result published in 2005 by Pisier.