Noncommutative de Leeuw theorems

TL;DR

This paper extends de Leeuw's restriction theorem to noncommutative group von Neumann algebras, establishing conditions under which Fourier multipliers on larger groups restrict to subgroups, with new approximation techniques.

Contribution

It introduces a noncommutative version of de Leeuw's theorem for a broad class of group pairs, using novel approximation methods replacing Gaussian dilations.

Findings

Lp-bounded Fourier multipliers restrict from G to H under certain conditions

New estimates on almost multiplicative maps are developed

Compactification and periodization results are obtained

Abstract

Let H be a subgroup of some locally compact group G. Assume H is approximable by discrete subgroups and G admits neighborhood bases which are "almost-invariant" under conjugation by finite subsets of H. Let $m: G \to \mathbb{C}$ be a bounded continuous symbol giving rise to an Lp-bounded Fourier multiplier (not necessarily cb-bounded) on the group von Neumann algebra of G for some $1 \le p \le \infty$. Then, $m_{\mid_H}$ yields an Lp-bounded Fourier multiplier on the group von Neumann algebra of H provided the modular function $\Delta_H$ coincides with $\Delta_G$ over H. This is a noncommutative form of de Leeuw's restriction theorem for a large class of pairs (G,H), our assumptions on H are quite natural and recover the classical result. The main difference with de Leeuw's original proof is that we replace dilations of gaussians by other approximations of the identity for which certain new estimates on almost multiplicative maps are crucial. Compactification via lattice approximation and periodization theorems are also investigated.