A Mikhlin multiplier theory for free groups and amalgamated free products of von Neumann algebras
A Mikhlin multiplier theory for free groups and amalgamated free products of von Neumann algebras
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DOI：
10.1016/j.aim.2022.108394
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发表时间：
2022-07
影响因子：
1.7
通讯作者：
T. Mei;Éric Ricard;Quanhua Xu
中科院分区：
文献类型：
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作者：
T. Mei;Éric Ricard;Quanhua Xu
得到了非交换自由群的Mikhlin乘子理论。设F∞是无限多个生成元{g1，g2，⋯}上的自由群。给定d≥1和Zd上的一个满足经典Mikhlin条件的有界符号m，由λ(G)↦m(k 1，⋯，k d)λ(G)定义的线性映射Mm：C[F∞]→C[F∞]对于所有1<；p<，g=gi 1 k 1λg i n k n⋯F∈F∞(其中k L在m(k 1，⋯，k d)中kˆ∞=0)，推广到L p(F)上的完全有界映射；∞，其中Fˆ∞是F∞的群von Neumann代数。在此过程中，我们建立了一个平台，将von Neumann代数张量积上的L p-完全有界映射转换到相应的合并自由积上的L p-完全有界映射。类似的结果也适用于任何离散群的自由积。
We obtain a Mikhlin multiplier theory for the nonabelian free groups. Let F∞ be a free group on infinite many generators {g 1, g 2,⋯}. Given d≥ 1 and a bounded symbol m on Z d satisfying the classical Mikhlin condition, the linear map M m: C [F∞]→ C [F∞] defined by λ (g)↦ m (k 1,⋯, k d) λ (g) for g= g i 1 k 1⋯ g i n k n∈ F∞ in reduced form (with k l= 0 in m (k 1,⋯, k d) for l> n), extends to a completely bounded map on L p (F ˆ∞) for all 1< p<∞, where F ˆ∞ is the group von Neumann algebra of F∞. In the process, 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. A similar result holds for any free product of discrete groups.