Spatial models of Boolean actions and groups of isometries

Kwiatkowska, Aleksandra; Solecki, Sławomir

doi:10.1017/s0143385709001138

Cited by 7 publications

(8 citation statements)

References 7 publications

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“…near-action of G has a Borel spatial model. This result was recently extended by Kwiatkowska and Solecki [13] to apply to all isometry groups of locally compact metric spaces.…”

Section: Again It Is Part Of the Definition Of A Near-action That η(mentioning

confidence: 83%

On the pointwise implementation of near-actions

Törnquist

2011

Trans. Amer. Math. Soc.

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“…near-action of G has a Borel spatial model. This result was recently extended by Kwiatkowska and Solecki [13] to apply to all isometry groups of locally compact metric spaces.…”

Section: Again It Is Part Of the Definition Of A Near-action That η(mentioning

confidence: 83%

On the pointwise implementation of near-actions

Törnquist

2011

Trans. Amer. Math. Soc.

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“…Kwiatkowska and the second author extended this and Mackey's result, proving that Boolean actions of isometry groups of locally compact separable metric spaces have spatial models [6].…”

Section: Introductionmentioning

confidence: 82%

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

Moore

Solecki

2012

Journal of Functional Analysis

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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 (R N , γ ∞ ) which is in contrast with the Cameron-Martin Theorem.

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“…THEOREM 2.5. [KS11] Every probability measure preserving Boolean action of the isometry group of a separable locally compact metric space has a spatial realization.…”

Section: Proof Of Theorem 13 (A)mentioning

confidence: 99%

On Polish groups admitting non-essentially countable actions

Kechris¹,

Malicki²,

Panagiotopoulos

et al. 2020

Ergod. Th. Dynam. Sys.

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It is a long-standing open question whether every Polish group that is not locally compact admits a Borel action on a standard Borel space whose associated orbit equivalence relation is not essentially countable. We answer this question positively for the class of all Polish groups that embed in the isometry group of a locally compact metric space. This class contains all non-archimedean Polish groups, for which we provide an alternative proof based on a new criterion for non-essential countability. Finally, we provide the following variant of a theorem of Solecki: every infinite-dimensional Banach space has a continuous action whose orbit equivalence relation is Borel but not essentially countable.

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Spatial models of Boolean actions and groups of isometries

Cited by 7 publications

References 7 publications

On the pointwise implementation of near-actions

On the pointwise implementation of near-actions

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

On Polish groups admitting non-essentially countable actions

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