Mathematics – Functional Analysis
Scientific paper
2012-01-18
Mathematics
Functional Analysis
12 pages. Minor changes in the introduction/abstract. Argument in section 3 is streamlined. Question in concluding remarks of
Scientific paper
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.
Moore Justin Tatch
Solecki Slawomir
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