## 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 the infinite-dimensional Gaussian measure space (RN,γ∞) which is in contrast with the Cameron-Martin Theorem.

Original language | English (US) |
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Pages (from-to) | 3224-3234 |

Number of pages | 11 |

Journal | Journal of Functional Analysis |

Volume | 263 |

Issue number | 10 |

DOIs | |

State | Published - Nov 15 2012 |

Externally published | Yes |

## Keywords

- Boolean action
- Cameron-Martin
- Group action
- Infinite-dimensional
- Point realization
- Polish group
- Spatial model
- Whirly

## ASJC Scopus subject areas

- Analysis