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

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 $T_m:L_p(\mathcal{L} G) \rightarrow L_p(\mathcal{L} G)$ 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 $\mathrm{Id} \rtimes T_m: L_p(\mathcal{M} \rtimes_\theta G) \rightarrow L_p(\mathcal{M} \rtimes_\theta G)$ is also completely bounded and that $$\| \mathrm{Id} \rtimes T_m: L_p(\mathcal{M} \rtimes_\theta G) \rightarrow L_p(\mathcal{M} \rtimes_\theta G) \|_{\mathrm{cb}} \leq \| T_m \|_{\mathrm{cb}}$$ provided that $\theta$ is amenable and trace-preserving. Furthermore, our construction allow to extend $G$-equivariant completely bounded operators $S: L_p(\mathcal{M}) \rightarrow L_p(\mathcal{M})$ to the crossed-product, so that $$\|S \rtimes \mathrm{Id}: L_p(\mathcal{M} \rtimes_\theta G) \rightarrow L_p(\mathcal{M} \rtimes_\theta G) \|_{\mathrm{cb}} \leq C^\frac{1}{p} \, \| S \|_{\mathrm{cb}}$$ whenever $\theta$ is trace-preserving, amenable and its generalized F{\o}lner sets satisfy certain accretivity property measured by the constant $1 \leq C$. 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 H\"omander-Mikhlin theorem.