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

Abstract

Let $Γ\curvearrowright Ω$ be a measure-preserving action and $\mathcal{L} Γ\hookrightarrow L^\infty(Ω) \rtimes Γ$ the natural inclusion of the group von Neumann algebra into the crossed product. When $μ(Ω) = \infty$, 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 $Ω$ has an invariant mean there is an isometric embedding of $L^p(\mathcal{L} Γ)$ into an ultrapower of $L^p(Ω\rtimes Γ)$ that intertwines Fourier multipliers and is $\mathcal{L} Γ$-bimodular. As a consequence we obtain the lower transference bound \[ \big\| T_m: L^p(\mathcal{L} Γ) \to L^p(\mathcal{L} Γ) \big\| \leq \big\| (\mathrm{id} \rtimes T_m): L^p(Ω\rtimes Γ) \to L^p(Ω\rtimes Γ) \big\|, \] and the same follows for complete norms. The condition of having an invariant mean is quite restrictive. Therefore, we explore whether other equivariant embeddings $Φ: \mathcal{L} Γ\to L^\infty(Ω)$ yield a more general transference result. We show that the transference proof above works verbatim whenever $Φ$ 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.