Lower bounds in $L^p$-transference for crossed-products

Let $\Gamma \curvearrowright \Omega$ be a measure-preserving action and $\mathcal{L} \Gamma \hookrightarrow L^\infty(\Omega) \rtimes \Gamma$ the natural inclusion of the group von Neumann algebra into the crossed product. When $\mu(\Omega) = \infty$, we have that this natural embedding is not trace-preserving and therefore does not extends boundedly to the associated noncommutative $L^p$-spaces. Nevertheless, we show that when $\Omega$ has an invariant mean there is an isometric embedding of $L^p(\mathcal{L} \Gamma)$ into an ultrapower of $L^p(\Omega \rtimes \Gamma)$ that intertwines Fourier multipliers and is $\mathcal{L} \Gamma$-bimodular. As a consequence we obtain the lower transference bound $$\big\| T_m: L^p(\mathcal{L} \Gamma) \to L^p(\mathcal{L} \Gamma) \big\| \leq \big\| (\mathrm{id} \rtimes T_m): L^p(\Omega \rtimes \Gamma) \to L^p(\Omega \rtimes \Gamma) \big\|,$$ and the same follows for complete norms. The condition of having an invariant mean is quite restrictive. Therefore, we explore whether other equivariant embeddings $\Phi: \mathcal{L} \Gamma \to L^\infty(\Omega)$ yield a more general transference result. We show that the transference proof above works verbatim whenever $\Phi$ is completely positive, amenable (in the sense of inducing an amenable correspondence) and intertwines Fourier multipliers at the $L^2$-level. Although no new transference results are obtained, both the classification of equivariant maps and the study their amenability may be of independent interest to some readers.