  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.
