自由群和von Neumann algebras的混合自由积的Hörmander-Mikhlin乘子理论
A Hörmander-Mikhlin multiplier theory for free groups and amalgamated free products of von Neumann algebras
摘要：
我们建立了一个平台，将vonNeumann代数张量积上的$L_p$-完全有界映射转化为相应的合并自由积上的$L_p$ -完全有界映射。作为结果，我们得到了群的自由积的Hörmander-Mikhlin乘子理论.设$\mathbb{F}_\infty$是无限生成元$\{g_1, g_2,\cdots\}$上的自由群.给定$d\ge1$和满足经典Hörmander-Mikhlin条件的$\mathbb{Z}^d$上的有界符号$m$，由$λ(g)\mapsto m(k_1,\cdots, k_d)λ(g)$为$g=g_{i_1}^{k_1}\cdots g_{i_n}^{k_n}\in\mathbb{F}_\infty$定义的线性映射$M_m:\mathbb{C}[\mathbb{F}_\infty]\to \mathbb{C}[\mathbb{F}_\infty]$（$k_l=0$在$l>n$的$m(k_1,\cdots, k_d)$中），对于所有$1<p<\infty$，扩展到$L_p(\widehat{\mathbb{F}}_\infty)$上的完全有界映射，其中$\widehat{\mathbb{F}}_\infty$是$\mathbb{F}_\infty$的群von Neumann代数。类似的结果适用于离散群的任意自由积。
英文摘要：
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 $d\ge1$ 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 $λ(g)\mapsto m(k_1,\cdots, k_d)λ(g)$ 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.
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