Ergodic Multiplier Properties

TL;DR

This paper investigates properties ensuring that the product of certain ergodic actions remains ergodic, proves a multiplier theorem for locally compact abelian groups, and explores weak mixing properties in specific group actions.

Contribution

It introduces a multiplier theorem for locally compact abelian groups and characterizes weak mixing in actions of Moore and minimally weakly mixing groups.

Findings

Product of ergodic actions can be ergodic under certain conditions.

Multiplier theorem established for locally compact abelian groups.

Gaussian action of the Heisenberg group is weakly mixing but not mildly mixing.

Abstract

In this article we will see some properties that guarantee that a product of an ergodic non-singular action and a probability preserving ergodic action is also an ergodic action. We will start by proving 'The multiplier theorem' for locally compact abelian groups. Then we will show that for certain locally compact Polish groups (Moore groups, and minimally weakly mixing groups), a non-singular G action is weakly mixing if and only if any finite dimensional G-invariant subspace of L_\infty is trivial. Finally, we will show that the Gaussian action associated to the infinite dimensional irreducible representation of the continuous Heisenberg group, H_3, is weakly mixing but not mildly mixing.