A Hörmander-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 $1<p<\infty$, 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.