Representations of dynamical systems on Banach spaces not containing $l_{1}$

Glasner, Eli; Megrelishvili, Michael

doi:10.1090/s0002-9947-2012-05549-8

Cited by 39 publications

(107 citation statements)

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“…For compact X and (pseudo)metric space (Y, d) it is equivalent to the fragmentability. If X is Polish and (Y, d) is a separable metric space then f : X → Y is fragmented iff f is a Baire class 1 function (i.e., the inverse image under f of every open set is F σ ), [10,14].…”

Section: Fragmented Mapsmentioning

confidence: 99%

“…Such families play a major role in the theory of tame dynamical systems. See, for example, [12,11,14,15,16].…”

Section: Fragmented Mapsmentioning

confidence: 99%

“…It is based on results of Rosenthal [26], Talagrand [29,Theorem 14.1.7] and van Dulst [5]. See also [14,Sect. 4].…”

Section: Fragmented Mapsmentioning

confidence: 99%

“…Its prototype is a well known fact that every monotonic function [a, b] → R is a Baire 1 function. [14,Lemma 2.3.3] it is enough to show that every map q i • f is fragmented. So we can assume that our order preserving function is of the form f : X → Y = [0, 1].…”

Section: Order Preserving Mapsmentioning

confidence: 99%

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A note on tameness of families having bounded variation

Megrelishvili¹

2017

Topology and its Applications

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Abstract. We show that for arbitrary linearly ordered set (X, ≤) any bounded family of (not necessarily, continuous) real valued functions on X with bounded total variation does not contain independent sequences. We obtain generalized Helly's sequential compactness type theorems. One of the theorems asserts that for every compact metric space (Y, d) the compact space BVr (X, Y ) of all functions X → Y with variation ≤ r is sequentially compact in the pointwise topology. Another Helly type theorem shows that the compact space M + (X, Y ) of all order preserving maps X → Y is sequentially compact where Y is a compact metrizable partially ordered space in the sense of Nachbin.

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Section: Fragmented Mapsmentioning

confidence: 99%

“…Such families play a major role in the theory of tame dynamical systems. See, for example, [12,11,14,15,16].…”

Section: Fragmented Mapsmentioning

confidence: 99%

“…It is based on results of Rosenthal [26], Talagrand [29,Theorem 14.1.7] and van Dulst [5]. See also [14,Sect. 4].…”

Section: Fragmented Mapsmentioning

confidence: 99%

Section: Order Preserving Mapsmentioning

confidence: 99%

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A note on tameness of families having bounded variation

Megrelishvili¹

2017

Topology and its Applications

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“…In [7], E. Glasner and M. Megrelishvili define a compact space to be weakly Radon-Nikodým (WRN for short) if and only if it is homeomorphic to a weak * -compact subset of the dual of a Banach space not containing an isomorphic copy of ℓ 1 . In [8], they ask whether the class of WRN compact spaces is stable under continuous images.…”

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confidence: 99%

On weakly Radon–Nikodým compact spaces

Martínez-Cervantes

2017

Isr. J. Math.

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Abstract. A compact space is said to be weakly Radon-Nikodým if it is homeomorphic to a weak * -compact subset of the dual of a Banach space not containing an isomorphic copy of ℓ 1 . In this paper we provide an example of a continuous image of a Radon-Nikodým compact space which is not weakly Radon-Nikodým. Moreover, we define a superclass of the continuous images of weakly Radon-Nikodým compact spaces and study its relation with Corson compacta and weakly Radon-Nikodým compacta.In [9], I. Namioka defined a compact space K to be Radon-Nikodým (RN for short) if and only if it is homeomorphic to a weak * -compact subset of a dual Banach space with the Radon-Nikodým property. One of the most important questions posed by Namioka in this paper was whether the class of RN compact spaces is closed under continuous images. This question was solved negatively by A. Avilés and P. Koszmider in [3].The class of weakly Radon-Nikodým compact spaces generalizes the class of RN compact spaces. In [7], E. Glasner and M. Megrelishvili define a compact space to be weakly Radon-Nikodým (WRN for short) if and only if it is homeomorphic to a weak * -compact subset of the dual of a Banach space not containing an isomorphic copy of ℓ 1 . In [8], they ask whether the class of WRN compact spaces is stable under continuous images. In these papers, a picture of this class of compact spaces is given and it is characterized in terms of Banach spaces of continuous functions. They prove that linearly ordered compact spaces are WRN. Moreover, [8] contains a proof by S. Todorcevic that βN is not WRN.In this work we answer negatively the question of E. Glasner and M. Megrelishvili by proving that a modification of the construction given in [3] provides an example of a continuous image of a RN compact space which is not WRN. We construct a RN compact space L 0 , a non-WRN compact space L 1 and a surjective continuous function π : L 0 −→ L 1 in the same way as in [3].In the second section we define quasi WRN compact spaces, a superclass of the continuous images of WRN compact spaces. We prove that zero-dimensional quasi WRN compact spaces are WRN. We also show other relations of quasi WRN compacta with Corson and Eberlein compacta, including an example of a Corson compact space which is not quasi WRN.

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Banach Representations and Affine Compactifications of Dynamical Systems

Glasner

Megrelishvili

2013

Asymptotic Geometric Analysis

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To every Banach space V we associate a compact right topological affine semigroup E(V ). We show that a separable Banach space V is Asplund if and only if E(V ) is metrizable, and it is Rosenthal (i.e. it does not contain an isomorphic copy of l 1 ) if and only if E(V ) is a Rosenthal compactum. We study representations of compact right topological semigroups in E(V ). In particular, representations of tame and HNS-semigroups arise naturally as enveloping semigroups of tame and HNS (hereditarily non-sensitive) dynamical systems, respectively. As an application we obtain a generalization of a theorem of R. Ellis. A main theme of our investigation is the relationship between the enveloping semigroup of a dynamical system X and the enveloping semigroup of its various affine compactifications Q(X). When the two coincide we say that the affine compactification Q(X) is E-compatible. This is a refinement of the notion of injectivity. We show that distal non-equicontinuous systems do not admit any E-compatible compactification. We present several new examples of non-injective dynamical systems and examine the relationship between injectivity and E-compatibility.1.1. Semigroups and actions. Let S be a semigroup which is also a topological space. By λ a : S → S, x → ax and ρ a : S → S, x → xa we denote the left and right a-transitions. The subset Λ(S) := {a ∈ S : λ a is continuous} is called the topological center of S. Definition 1.1. A semigroup S as above is said to be:(1) a right topological semigroup if every ρ a is continuous.(Let A be a subsemigroup of a right topological semigroup S. If A ⊂ Λ(S) then the closure cl(A) is a right topological semigroup. In general, cl(A) is not necessarily a subsemigroup of S (even if S is compact right topological and A is a left ideal). Also Λ(S) may be empty for general compact right topological semigroup S. See [7, p. 29].Definition 1.2. Let S be a semitopological semigroup with a neutral element e. Let π : S × X → X be a left action of S on a topological space X. This means that ex = x and s 1 (s 2 x) = (s 1 s 2 )x for all s 1 , s 2 ∈ S and x ∈ X, where as usual, we write sx instead of π(s, x) = λ s (We say that X is a dynamical S-system (or an S-space or an S-flow ) if the action π is separately continuous (that is, if all orbit maps ρ x : S → X and all translations λ s : X → X are continuous). We sometimes write it as a pair (S, X).A right system (X, S) can be defined analogously. If S op is the opposite semigroup of S with the same topology then (X, S) can be treated as a left system (S op , X) (and vice versa). Fact 1.3. [43]Let G be aČech-complete (e.g., locally compact or completely metrizable) semitopological group. Then every separately continuous action of G on a compact space X is continuous.Notation: All semigroups S are assumed to be monoids, i.e, semigroups with a neutral element which will be denoted by e. Also actions are monoidal (meaning ex = x, ∀x ∈ X) and separately continuous. We reserve the symbol G for the case when S is a group. All right topological semig...

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Representations of dynamical systems on Banach spaces not containing $l_{1}$

Cited by 39 publications

References 41 publications

A note on tameness of families having bounded variation

A note on tameness of families having bounded variation

On weakly Radon–Nikodým compact spaces

Banach Representations and Affine Compactifications of Dynamical Systems

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