A Boolean action of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:mi>C</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>M</mml:mi><mml:mo>,</mml:mo><mml:mi>U</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:math> without a spatial model and a re-examination of the Cameron–Martin Theorem

Moore, Justin Tatch; Solecki, Sławomir

doi:10.1016/j.jfa.2012.08.004

Cited by 3 publications

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“…In the countable theory, a distinction is made between an action which is defined everywhere on the underlying set and a near-action which is only defined almost everywhere, e.g., see [48]. This distinction is non-trivial, since there are natural examples of systems of countable complexity for Polish group near-actions that cannot be realized by a pointwise action, e.g., see [20, 39].…”

Section: Introductionmentioning

confidence: 99%

An uncountable Furstenberg–Zimmer structure theory

Jamneshan

2022

Ergod. Th. Dynam. Sys.

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Furstenberg–Zimmer structure theory refers to the extension of the dichotomy between the compact and weakly mixing parts of a measure-preserving dynamical system and the algebraic and geometric descriptions of such parts to a conditional setting, where such dichotomy is established relative to a factor and conditional analogs of those algebraic and geometric descriptions are sought. Although the unconditional dichotomy and the characterizations are known for arbitrary systems, the relative situation is understood under certain countability and separability hypotheses on the underlying groups and spaces. The aim of this article is to remove these restrictions in the relative situation and establish a Furstenberg–Zimmer structure theory in full generality. As an independent byproduct, we establish a connection between the relative analysis of systems in ergodic theory and the internal logic in certain Boolean topoi.

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Section: Introductionmentioning

confidence: 99%

An uncountable Furstenberg–Zimmer structure theory

Jamneshan

2022

Ergod. Th. Dynam. Sys.

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Unitary representations of the groups of measurable and continuous functions with values in the circle

Solecki

2014

Journal of Functional Analysis

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Abstract. We give a classification of unitary representations of certain Polish, not necessarily locally compact, groups: the groups of all measurable functions with values in the circle and the groups of all continuous functions on compact, second countable, zero-dimensional spaces with values in the circle. In the proofs of our classification results, certain structure theorems and factorization theorems for linear operators are used. IntroductionWe study unitary representations of the groups of measurable and continuous functions with values in the circle. A description of unitary representations of such groups is of interest especially in view of recent considerable activity around topological and measurable dynamics of these groups; see the remarks below. The reader may consult [14] for background information on dynamics of Polish non-locally compact groups. (Recall that a topological group is Polish if its topology is metrizable by a complete separable metric.)For a Borel probability measure µ on a standard Borel space and a topological group H, let L 0 (µ, H) be the topological group of all µ-equivalence classes of µ-measurable functions with values in H. The multiplication on L 0 (µ, H) is implemented pointwise and the topology is the convergence in measure topology. Groups of this form were perhaps first systematically considered in [3] to provide an embedding of each topological group into a connected group. Recently, the more particular Polish groups L 0 (µ, T) generated substantial interest in the context of extreme amenability

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