An Operator-Valued Haagerup Inequality for Hyperbolic Groups
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866915916774637568 |
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| author | Toyota, Ryo Yang, Zhiyuan |
| author_facet | Toyota, Ryo Yang, Zhiyuan |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2311_13651 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | An Operator-Valued Haagerup Inequality for Hyperbolic Groups Toyota, Ryo Yang, Zhiyuan Operator Algebras Functional Analysis Group Theory 47 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. |
| title | An Operator-Valued Haagerup Inequality for Hyperbolic Groups |
| topic | Operator Algebras Functional Analysis Group Theory 47 |
| url | https://arxiv.org/abs/2311.13651 |