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

Document Type: Article dedicated to Uffe Haagerup



We estimate the norms of many matrix coefficients of irreducible uniformly bounded representations of ${\rm SL}(2,\mathbb{R})$ 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.