Relatively finite measure-preserving extensions and lifting multipliers by Rokhlin cocycles

TL;DR

This paper investigates conditions under which measure-preserving extensions with Rokhlin cocycles preserve the multiplier property, focusing on cases with countably many eigenvalues, and extends finite-rank module theory to non-singular systems.

Contribution

It introduces new conditions for lifting the multiplier property in measure-preserving extensions using Rokhlin cocycles, expanding the theory to non-singular base systems.

Findings

Extensions with countably many eigenvalues preserve the multiplier property under ergodicity.

Analogues of finite-rank module results are established for non-singular systems.

Provides criteria for lifting multipliers in measure-preserving extensions.

Abstract

We show that under some natural ergodicity assumptions extensions given by Rokhlin cocycles lift the multiplier property if the associated locally compact group extension has only countably many L^\infty-eigenvalues. We make use of some analogs of basic results from the theory of finite-rank modules associated to an extension of measure-preserving systems in the setting of a non-singular base.