A Boolean action of C(M,U(1)) without a spatial model and a re-examination of the Cameron-Martin theorem

TL;DR

This paper constructs a measure-preserving Boolean action of a function group on a probability algebra that cannot be realized pointwise, challenging classical realization theorems and re-examining the Cameron-Martin theorem in infinite dimensions.

Contribution

It demonstrates a new type of Boolean action without a point realization and provides a counterexample to the Cameron-Martin theorem in infinite-dimensional Gaussian spaces.

Findings

Existence of a measure-preserving Boolean action without point realization

Counterexample to the Cameron-Martin theorem in infinite dimensions

Introduction of the whirliness property in ergodic actions

Abstract

We will demonstrate that if M is an uncountable compact metric space, then there is an action of the Polish group of all continuous functions from M to U(1) on a separable probability algebra which preserves the measure and yet does not admit a point realization in the sense of Mackey. This is achieved by exhibiting a strong form of ergodicity of the Boolean action known as whirliness. This is in contrast with Mackey's point realization theorem, which asserts that any measure preserving Boolean action of a locally compact second countable group on a separable probability algebra can be realized as an action on the points of the associated probability space. In the course of proving the main theorem, we will prove a result concerning infinite dimensional Gaussian measure space which is in contrast with the Cameron-Martin Theorem.