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Let $\Gamma \curvearrowright \Omega$ be a measure-preserving action and$\mathcal{L} \Gamma \hookrightarrow L^\infty(\Omega) \rtimes \Gamma$ thenatural inclusion of the group von Neumann algebra into the crossed product.When $\mu(\Omega) = \infty$, we have that this natural embedding is nottrace-preserving and therefore does not extends boundedly to the associatednoncommutative $L^p$-spaces. Nevertheless, we show that when $\Omega$ has aninvariant mean there is an isometric embedding of $L^p(\mathcal{L} \Gamma)$into an ultrapower of $L^p(\Omega \rtimes \Gamma)$ that intertwines Fouriermultipliers and is $\mathcal{L} \Gamma$-bimodular. As a consequence we obtainthe 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, weexplore whether other equivariant embeddings $\Phi: \mathcal{L} \Gamma \toL^\infty(\Omega)$ yield a more general transference result. We show that thetransference proof above works verbatim whenever $\Phi$ is completely positive,amenable (in the sense of inducing an amenable correspondence) and intertwinesFourier multipliers at the $L^2$-level. Although no new transference resultsare obtained, both the classification of equivariant maps and the study theiramenability may be of independent interest to some readers.

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