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

TL;DR

This paper investigates lower bounds in $L^p$-transference for crossed-product algebras, establishing conditions under which embeddings and Fourier multiplier bounds can be transferred between group von Neumann algebras and crossed products.

Contribution

It introduces an isometric embedding of $L^p( ext{group von Neumann algebra})$ into ultrapowers of crossed products under invariant mean conditions, and explores broader conditions for transference via equivariant maps.

Findings

Established lower transference bounds for Fourier multipliers.

Constructed isometric embeddings under invariant mean assumptions.

Analyzed conditions for general transference results beyond invariant mean.

Abstract

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.