Non-Commutative Wiener-Wintner theorem for amenable group actions

Abstract

Let $G$ be a locally compact, second countable, amenable group acting on a finite von Neumann algebra $(\mathcal{M},\tau )$ by trace-preserving automorphisms. In this article, we establish a Jacobs-de Leeuw-Glicksberg decomposition for this action, yielding a decomposition of $\mathcal{M}$ into its almost periodic and weakly mixing components. We also prove a noncommutative version of the van der Corput lemma. As an application, we establish a noncommutative Wiener-Wintner theorem for amenable group actions on finite von Neumann algebras.

31 pages, proved the Wiener-Wintner theorem for all amenable group actions