An Operator-Valued Haagerup Inequality for Hyperbolic Groups

Ryo Toyota; Zhiyuan Yang

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arXiv · 2311.13651

An Operator-Valued Haagerup Inequality for Hyperbolic Groups

Ryo Toyota·Zhiyuan Yang

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

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Ryo Toyota, Zhiyuan Yang. 2023-11-22. An Operator-Valued Haagerup Inequality for Hyperbolic Groups. https://arxiv.org/abs/2311.13651

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