Crossed-Product Extensions of L_p-Bounds for Amenable Actions

2016/11/25 by Adrián M. González‐Pérez, González-Pérez, A. M.
Mathematics · Physics and Astronomy · #Advanced Operator Algebra Research #Noncommutative and Quantum Gravity Theories #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1611.08486

Abstract

We will extend earlier transference results of Neuwirth and Ricard from the context of noncommutative L_p-spaces associated with amenable groups to that of noncommutative L_p-spaces over crossed products of amenable and trace-preserving actions. Namely, if T_m:L_p(L G) → L_p(L G) is a completely bounded Fourier multiplier, where L G ⊂ B(L₂ G) is the von Neumann algebra of G, we will see that Id \rtimes T_m: L_p(M \rtimes_θG) → L_p(M \rtimes_θG) is also completely bounded and that ‖ Id \rtimes T_m: L_p(M \rtimes_θG) → L_p(M \rtimes_θG) ‖_cb ≤ ‖ T_m ‖_cb provided that θ is amenable and trace-preserving. Furthermore, our construction allow to extend G-equivariant completely bounded operators S: L_p(M) → L_p(M) to the crossed-product, so that ‖S \rtimes Id: L_p(M \rtimes_θG) → L_p(M \rtimes_θG) ‖_cb ≤ C^(1)/(p) ‖ S ‖_cb whenever θ is trace-preserving, amenable and its generalized Følner sets satisfy certain accretivity property measured by the constant 1 ≤ C. As a corollary, we will obtain stability results for maximal L_p-bounds over crossed products. Such results imply the stability of certain assumptions recently used to prove a noncommutative generalization of the spectral Hömander-Mikhlin theorem.

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