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

Tao Mei (1) , Éric Ricard (2) , Quanhua Xu (3)

1 Department of Mathematics, Baylor University
2 LMNO - Laboratoire de Mathématiques Nicolas Oresme
3 LMB - Laboratoire de Mathématiques de Besançon (UMR 6623)

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

Domaines: Analyse fonctionnelle [math.FA]

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⟩