An Operator-Valued Haagerup Inequality for Hyperbolic Groups

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 \((k+1)\|f\|_2\). 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.