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

2021/03/07 by Tao Mei, Mei, Tao, Éric Ricard +3 · 1 citation
Mathematics · #46L50. Secondary: 46L52 #46L54 #Advanced Operator Algebra Research #Algebraic structures and combinatorial models #FOS: Mathematics #Functional Analysis (math.FA) #Geometric and Algebraic Topology #Operator Algebras (math.OA) #Primary: 46L07

paper · pdf · doi:10.48550/arxiv.2103.04368

openalex publication_date 2021/03/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 \mathbbF_∞ be a free group on infinite generators \g₁, g₂,⋯\. Given d≥1 and a bounded symbol m on ℤ^d satisfying the classical Hörmander-Mikhlin condition, the linear map M_m:ℂ[\mathbbF_∞]→ ℂ[\mathbbF_∞] defined by λ(g)↦ m(k₁,⋯, k_d)λ(g) for g=g_i1^k₁⋯ g_in^k_n∈\mathbbF_∞ in reduced form (with k_l=0 in m(k₁,⋯, k_d) for l>n), extends to a complete bounded map on L_p(\widehat\mathbbF_∞) for all 1

Citations

• Multipliers of the Fourier Algebras of Some Simple Lie Groups and Their Discrete Subgroups

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• The impact of Schur multipliers in harmonic analysis and operator algebras

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