Uniformly bounded representations and completely bounded multipliers of $\mathrm {SL}(2,\mathbb{R})$

Abstract

We estimate the norms of many matrix coefficients of irreducible uniformly bounded representations of $\mathrm {SL}(2,\mathbb{R})$ as completely bounded multipliers of the Fourier algebra. Our results suggest that the known inequality relating the uniformly bounded norm of a representation and the completely bounded norm of its coefficients may not be optimal.