Crossed-Product Extensions of $L_p$-Bounds for Amenable Actions

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 $T_m:L_p(\mathcal{L}G)\to L_p(\mathcal{L}G)$ is a completely bounded Fourier multiplier, where $\mathcal{L}G\subset \mathcal{B}(L_{2}G)$ is the von Neumann algebra of $G$, we will see that $\mathrm{Id}⋊T_m:L_p(\mathcal{M}{⋊}_{\theta }G)\to L_p(\mathcal{M}{⋊}_{\theta }G)$ is also completely bounded and that $$\Vert \mathrm{Id}⋊T_m:L_p(\mathcal{M}{⋊}_{\theta }G)\to L_p(\mathcal{M}{⋊}_{\theta }G){\Vert }_{\mathrm{cb}}\le \Vert T_m{\Vert }_{\mathrm{cb}}$$ provided that $\theta$ is amenable and trace-preserving. Furthermore, our construction allow to extend $G$-equivariant completely bounded operators $S:L_p(\mathcal{M})\to L_p(\mathcal{M})$ to the crossed-product, so that $$\Vert S⋊\mathrm{Id}:L_p(\mathcal{M}{⋊}_{\theta }G)\to L_p(\mathcal{M}{⋊}_{\theta }G){\Vert }_{\mathrm{cb}}\le C^{\frac{1}{p}}\Vert S{\Vert }_{\mathrm{cb}}$$ whenever $\theta$ is trace-preserving, amenable and its generalized Følner sets satisfy certain accretivity property measured by the constant $1\le C$. 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.