A H\"ormander-Mikhlin multiplier theory for free groups and amalgamated free products of von Neumann algebras

TL;DR

This paper develops a H"ormander-Mikhlin multiplier theory for free groups and amalgamated free products of von Neumann algebras, extending classical harmonic analysis results to noncommutative group von Neumann algebras.

Contribution

It introduces a transfer platform for $L_p$-completely bounded maps to free products, establishing a new multiplier theory for free group von Neumann algebras.

Findings

Established a H"ormander-Mikhlin multiplier theorem for free groups.

Extended the theory to amalgamated free products of von Neumann algebras.

Proved boundedness of certain multipliers on $L_p$ spaces for free groups.

Abstract

We establish a platform to transfer $L_p$-completely bounded maps on tensor products of von Neumann algebras to $L_p$-completely bounded maps on the corresponding amalgamated free products. As a consequence, we obtain a H\"ormander-Mikhlin multiplier theory for free products of groups. Let $\mathbb{F}_\infty$ be a free group on infinite generators $\{g_1, g_2,\cdots\}$. Given $d\ge1$ and a bounded symbol $m$ on $\mathbb{Z}^d$ satisfying the classical H\"ormander-Mikhlin condition, the linear map $M_m:\mathbb{C}[\mathbb{F}_\infty]\to \mathbb{C}[\mathbb{F}_\infty]$ defined by $\lambda(g)\mapsto m(k_1,\cdots, k_d)\lambda(g)$ for $g=g_{i_1}^{k_1}\cdots g_{i_n}^{k_n}\in\mathbb{F}_\infty$ in reduced form (with $k_l=0$ in $m(k_1,\cdots, k_d)$ for $l>n$), extends to a complete bounded map on $L_p(\widehat{\mathbb{F}}_\infty)$ for all $1<p<\infty$, where $\widehat{\mathbb{F}}_\infty$ is the group von Neumann algebra of $\mathbb{F}_\infty$. A similar result holds for any free product of discrete groups.