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 Hörmander-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 $d\ge 1$ and a bounded symbol $m$ on ${\mathbb{Z}}^d$ satisfying the classical Hörmander-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 $L_p({\hat{\mathbb{F}}}_{\infty })$ for all $1<p<\infty$, where ${\hat{\mathbb{F}}}_{\infty }$ is the group von Neumann algebra of ${\mathbb{F}}_{\infty }$. A similar result holds for any free product of discrete groups.

References in corpus (4)

• Simple Lie groups without the Approximation Property II
• Free Hilbert Transforms
• Approximation properties for noncommutative $L^p$-spaces of high rank lattices and nonembeddability of expanders
• Free Fourier Multipliers associated with the firstSegment