A H\"ormander-Mikhlin multiplier theory for free groups and amalgamated free products of von Neumann algebras
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
Domains
Functional Analysis [math.FA]Éric Ricard : Connect in order to contact the contributor
https://normandie-univ.hal.science/hal-03180793
Submitted on : Thursday, March 25, 2021-11:47:10 AM
Last modification on : Tuesday, October 17, 2023-11:36:03 AM
Dates and versions
Identifiers
- HAL Id : hal-03180793 , version 1
- ARXIV : 2103.04368
- DOI : 10.1016/j.aim.2022.108394
Cite
Tao Mei, Éric Ricard, Quanhua Xu. A H\"ormander-Mikhlin multiplier theory for free groups and amalgamated free products of von Neumann algebras. Advances in Mathematics, 2022, ⟨10.1016/j.aim.2022.108394⟩. ⟨hal-03180793⟩
Collections
36
View
0
Download