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

Justin Tatch Moore, Sławomir Solecki

Research output: Contribution to journalArticlepeer-review

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 languageEnglish (US)
Pages (from-to)3224-3234
Number of pages11
JournalJournal of Functional Analysis
Volume263
Issue number10
DOIs
StatePublished - Nov 15 2012

Keywords

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

ASJC Scopus subject areas

  • Analysis

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