A Mikhlin multiplier theory for free groups and amalgamated free products of von Neumann algebras

Abstract

We obtain a Mikhlin multiplier theory for the nonabelian free groups. Let F∞ be a free group on infinite many generators {g1,g2,⋯}. Given d≥1 and a bounded symbol m on Zd satisfying the classical Mikhlin condition, the linear map Mm:C[F∞]→C[F∞] defined by λ(g)↦m(k1,⋯,kd)λ(g) for g=gi1k1⋯ginkn∈F∞ in reduced form (with kl=0 in m(k1,⋯,kd) for l>n), extends to a completely bounded map on Lp(Fˆ∞) for all 1<p<∞, where Fˆ∞ is the group von Neumann algebra of F∞. In the process, we establish a platform to transfer Lp-completely bounded maps on tensor products of von Neumann algebras to Lp-completely bounded maps on the corresponding amalgamated free products. A similar result holds for any free product of discrete groups.