An Operator-Valued Haagerup Inequality for Hyperbolic Groups

Abstract

We study an operator-valued generalization of the Haagerup inequality for Gromov hyperbolic groups. In 1978, U. Haagerup showed that if $f$ is a function on the free group ${\mathbb{F}}_r$ which is supported on the $k$-sphere $S_k=\{x\in {\mathbb{F}}_r:\ell (x)=k\}$, then the operator norm of its left regular representation is bounded by $(k+1)\Vert f{\Vert }_2$. An operator-valued generalization of it was started by U. Haagerup and G. Pisier. One of the most complete form was given by A. Buchholz, where the ${\ell }^2$-norm in the original inequality was replaced by $k+1$ different matrix norms associated to word decompositions (this type of inequality is also called Khintchine-type inequality). We provide a generalization of Buchholz's result for hyperbolic groups.

7 pages