# 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
Document type :
Journal articles

https://hal-normandie-univ.archives-ouvertes.fr/hal-03180793
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Submitted on : Thursday, March 25, 2021 - 11:47:10 AM
Last modification on : Friday, April 22, 2022 - 1:24:26 PM

### Citation

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, Elsevier, 2022, ⟨10.1016/j.aim.2022.108394⟩. ⟨hal-03180793⟩

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