Hilbert transforms on graph products of finite von Neumann algebras
Abstract
We study Hilbert transforms on graph products of finite von Neumann algebras, with particular interest in their boundedness on the associated noncommutative Lp-spaces for 1<p<∞. We establish a generalized Cotlar identity for Hilbert transforms, valid on operators whose lengths exceed a constant depending only on the underlying graph. We further prove that graph products of finite von Neumann algebras satisfying a Haagerup-type inequality admit Lp-bounded Hilbert transforms, therefore extending the corresponding result of Mei and Ricard for free products of finite von Neumann algebras. In addition, we obtain several equivalent characterizations of this Haagerup-type inequality. Our results apply, in particular, to graph products of finite groups, right-angled Hecke von Neumann algebras, and graph products of finite quantum groups. As an application, we provide a complete characterization of a compactness problem posed by Ozawa for graph products of finite groups and right-angled Hecke von Neumann algebras.
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BibTeX
@misc{lu2026hilbert,
title = {{Hilbert transforms on graph products of finite von Neumann algebras}},
author = {Xiao-Qi Lu and Runlian Xia},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.00194},
eprint = {2607.00194},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.00194},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Hilbert transforms on graph products of finite von Neumann algebras
AU - Xiao-Qi Lu
AU - Runlian Xia
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2607.00194
UR - https://arxiv.org/abs/2607.00194
AB - We study Hilbert transforms on graph products of finite von Neumann algebras, with particular interest in their boundedness on the associated noncommutative L_p-spaces for 1<p<∞. We establish a generalized Cotlar identity for Hilbert transforms, valid on operators whose lengths exceed a constant depending only on the underlying graph. We further prove that graph products of finite von Neumann algebras satisfying a Haagerup-type inequality admit L_p-bounded Hilbert transforms, therefore extending the corresponding result of Mei and Ricard for free products of finite von Neumann algebras. In addition, we obtain several equivalent characterizations of this Haagerup-type inequality. Our results apply, in particular, to graph products of finite groups, right-angled Hecke von Neumann algebras, and graph products of finite quantum groups. As an application, we provide a complete characterization of a compactness problem posed by Ozawa for graph products of finite groups and right-angled Hecke von Neumann algebras.
ER -