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We will demonstrate that if M is an uncountable compact metric space, thenthere is an action of the Polish group of all continuous functions from M toU(1) on a separable probability algebra which preserves the measure and yetdoes not admit a point realization in the sense of Mackey. This is achieved byexhibiting a strong form of ergodicity of the Boolean action known aswhirliness. This is in contrast with Mackey's point realization theorem, whichasserts that any measure preserving Boolean action of a locally compact secondcountable group on a separable probability algebra can be realized as an actionon the points of the associated probability space. In the course of proving themain theorem, we will prove a result concerning infinite dimensional Gaussianmeasure space which is in contrast with the Cameron-Martin Theorem.

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