Compacts de Rosenthal

Godefroy, Gilles

doi:10.2140/pjm.1980.91.293

Cited by 55 publications

(37 citation statements)

References 5 publications

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“…(3.4), and since C D (K ) is also analytic, cf. [7], it is Borel. We shall show that, among AU-compacta, the compacta K A associated with Borel families A are the ones that can be embedded in B 1 (ω ω ) in a particularly simple way.…”

Section: The Johnson-lindenstrauss Spaces Jl(k )mentioning

confidence: 99%

“…[7]. Let us recall an important characterization of separable Rosenthal compacta due to Godefroy, introducing first some notation.…”

Section: The Spaces B 1 (S) and C D (K )mentioning

confidence: 99%

“…The space C D (K ) can be identified with the topological space (C(K ), τ D ), where τ D is the topology of pointwise convergence on D. Now, a separable compact space K is a Rosenthal compactum if, and only if, for any countable set D dense in K , the set C D (K ) is analytic, cf. [7,Theorem 4]. We shall also use a fundamental fact that Rosenthal compacta are Fréchet spaces, cf.…”

Section: The Spaces B 1 (S) and C D (K )mentioning

confidence: 99%

“…Following Godefroy [7], one can check that, for a separable compact space K , the measurable space (C(K ), Cyl(C(K ))) is standard if and only if, for each countable…”

Section: The Spaces B 1 (S) and C D (K )mentioning

confidence: 99%

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On Borel almost disjoint families

Marciszewski

Pol

2012

Monatsh Math

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There is a close correspondence between uncountable almost disjoint families of subsets of ω and Aleksandrov-Urysohn compacta (in short, AU-compacta)-separable, uncountable compact spaces whose second derived set is a singleton. We shall show in particular, that AU-compacta embeddable in the space of first Baire class functions on the Cantor set 2 ω , with the pointwise topology, are exactly the ones determined by almost disjoint families that are Borel sets in 2 ω , and they are also distinguished among AU-compacta by the property that the cylindrical σ -algebras of their function spaces are standard measurable spaces. Although the first condition implies the third one for arbitrary separable compact space, it is an open problem, whether the reverse implication is always true.

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Section: The Johnson-lindenstrauss Spaces Jl(k )mentioning

confidence: 99%

“…[7]. Let us recall an important characterization of separable Rosenthal compacta due to Godefroy, introducing first some notation.…”

Section: The Spaces B 1 (S) and C D (K )mentioning

confidence: 99%

Section: The Spaces B 1 (S) and C D (K )mentioning

confidence: 99%

“…Following Godefroy [7], one can check that, for a separable compact space K , the measurable space (C(K ), Cyl(C(K ))) is standard if and only if, for each countable…”

Section: The Spaces B 1 (S) and C D (K )mentioning

confidence: 99%

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On Borel almost disjoint families

Marciszewski

Pol

2012

Monatsh Math

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“…The fundamental facts about angelic spaces can be found in [5] . A remarkable class of angelic compact spaces, the Rosenthal compacts, has been studied by Godefroy in [6] .…”

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confidence: 99%

Some results in representable Banach spaces

Canela¹

1988

Publicacions Matemàtiques

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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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Compacts de Rosenthal

Cited by 55 publications

References 5 publications

On Borel almost disjoint families

On Borel almost disjoint families

Some results in representable Banach spaces

Banach Representations and Affine Compactifications of Dynamical Systems

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