Hilbert transforms on graph products of finite von Neumann algebras

Xiao-Qi Lu; Runlian Xia

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arXiv · 2607.00194

Hilbert transforms on graph products of finite von Neumann algebras

Xiao-Qi Lu·Runlian Xia

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Abstract

We study Hilbert transforms on graph products of finite von Neumann algebras, with particular interests on their boundedness on the associated noncommutative $L_p$-spaces for $1<p<\infty$. 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 and show, in particular, that it is equivalent to the graph product being generated by finite-dimensional von Neumann algebras with uniformly bounded dimensions. 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 positive answers to a compactness problem posed by Ozawa in the setting of graph products of finite groups and right-angled Hecke von Neumann algebras.

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BibTeX RIS

Xiao-Qi Lu, Runlian Xia. 2026-06-30. Hilbert transforms on graph products of finite von Neumann algebras. https://arxiv.org/abs/2607.00194

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