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

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 ${\mathbb{F}}_{\mathrm{\infty }}$ be a free group on infinite generators ${g_1, g_2,\cdots}$\mathrm{.}Given$d\ge1$andaboundedsymbol$m$on$$\mathbb{Z}$^d$satisfyingtheclassicalH\ddot{o}rmander-Mikhlincondition,thelinearmap$M_m:$\mathbb{C}$[$\mathbb{F}$\infty]\to $\mathbb{C}$[$\mathbb{F}$\infty]$definedby$\lambda(g)\mapsto m(k_1,\cdots, k_d)\lambda(g)$for$g=g_{i_1}^{k_1}\cdots g_{i_n}^{k_n}\in$\mathbb{F}$\infty$inreducedform(with$k_l=0$in$m(k_1,\cdots, k_d)$for$l>n$),extendstoacompleteboundedmapon$L_p(\widehat{$\mathbb{F}$}\infty)$forall$1<p<\infty$,where$\widehat{$\mathbb{F}$}\infty$isthegroupvonNeumannalgebraof$$\mathbb{F}$\infty$. A similar result holds for any free product of discrete groups.