A Boolean action of without a spatial model and a re-examination of the Cameron-Martin Theorem

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• 作者：Justin Tatch Moore ; S?awomir Solecki
• 关键词：Boolean action ; Cameron&ndash ; Martin ; Group action ; Infinite-dimensional ; Point realization ; Polish group ; Spatial model ; Whirly
• 刊名：Journal of Functional Analysis
• 出版年：2012
• 出版时间：15 November, 2012
• 年：2012
• 卷：263
• 期：10
• 页码：3224-3234
• 全文大小：143 K

文摘

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 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 which is in contrast with the Cameron-Martin Theorem.