Commutator criteria for strong mixing II. More general and simpler

TL;DR

This paper introduces a new, more general commutator-based criterion for establishing the strong mixing property in unitary representations of topological groups, unifying previous results and simplifying proofs.

Contribution

It generalizes and simplifies the criteria for strong mixing, applying to both discrete and continuous flows, and offers an alternative proof for the strong mixing of regular representations.

Findings

Unified criterion for strong mixing in unitary representations

Applicable to discrete and continuous flows

Simplified proof of regular representation mixing

Abstract

We present a new criterion, based on commutator methods, for the strong mixing property of unitary representations of topological groups equipped with a proper length function. Our result generalises and unifies recent results on the strong mixing property of discrete flows $\{U^N\}_{N\in\mathbb Z}$ and continuous flows $\{{\rm e}^{-itH}\}_{t\in\mathbb R}$ induced by unitary operators $U$ and self-adjoint operators $H$ in a Hilbert space. As an application, we present a short alternative proof (not using convolutions) of the strong mixing property of the left regular representation of $\sigma$-compact locally compact groups.