Some remarks on universality properties of $ℓ_∞/c₀$

Mikołaj Krupski; Witold Marciszewski

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Displaying similar documents to “Some remarks on universality properties of $ℓ_{\infty} / c ₀$ ”

A condition equivalent to uniform ergodicity

Maria Elena Becker (2005)

Studia Mathematica

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Let T be a linear operator on a Banach space X with $s u p ₙ | | T ⁿ / n^{w} | | < \infty$ for some 0 ≤ w < 1. We show that the following conditions are equivalent: (i) $n^{- 1} \sum_{k = 0}^{n - 1} T^{k}$ converges uniformly; (ii) $c l (I - T) X = z \in X : l i m_{n} \sum_{k = 1}^{n} T^{k} z / k e x i s t s$ .

Reflecting character and pseudocharacter

Lucia R. Junqueira, Alberto M. E. Levi (2015)

Commentationes Mathematicae Universitatis Carolinae

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We say that a cardinal function $φ$ reflects an infinite cardinal $κ$ , if given a topological space $X$ with $φ (X) \geq κ$ , there exists $Y \in {[X]}^{\leq κ}$ with $φ (Y) \geq κ$ . We investigate some problems, discussed by Hodel and Vaughan in Reflection theorems for cardinal functions, Topology Appl. 100 (2000), 47–66, and Juhász in Cardinal functions and reflection, Topology Atlas Preprint no. 445, 2000, related to the reflection for the cardinal functions character and pseudocharacter. Among other results, we present some new equivalences...

Cardinal sequences of length < ω₂ under GCH

István Juhász, Lajos Soukup, William Weiss (2006)

Fundamenta Mathematicae

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Let (α) denote the class of all cardinal sequences of length α associated with compact scattered spaces (or equivalently, superatomic Boolean algebras). Also put $_{λ} (α) = s \in (α) : s (0) = λ = m i n [s (β) : β < α]$ . We show that f ∈ (α) iff for some natural number n there are infinite cardinals $λ ₀ i > λ ₁ > . . . > λ_{n - 1}$ and ordinals $α ₀, . . ., α_{n - 1}$ such that $α = α ₀ + \dots + α_{n - 1}$ and $f = f ₀ ⏜ f ₁ ⏜ . . . ⏜ f_{n - 1}$ where each $f_{i} \in_{λ_{i}} (α_{i})$ . Under GCH we prove that if α < ω₂ then (i) $_{ω} (α) = s \in^{α} ω, ω ₁ : s (0) = ω$ ; (ii) if λ > cf(λ) = ω, $_{λ} (α) = s \in^{α} λ, λ ⁺ : s (0) = λ, s^{- 1} λ i s ω ₁ - c l o s e d i n α$ ; (iii) if cf(λ) = ω₁, $_{λ} (α) = s \in^{α} λ, λ ⁺ : s (0) = λ, s^{- 1} λ i s ω - c l o s e d a n d s u c c e s s o r - c l o s e d i n α$ ; (iv) if cf(λ) > ω₁, $_{λ} (α) =^{α} λ$ . This yields a complete characterization of the classes (α) for all...

On ordinals accessible by infinitary languages

Saharon Shelah, Pauli Väisänen, Jouko Väänänen (2005)

Fundamenta Mathematicae

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Let λ be an infinite cardinal number. The ordinal number δ(λ) is the least ordinal γ such that if ϕ is any sentence of $L_{λ ⁺ ω}$ , with a unary predicate D and a binary predicate ≺, and ϕ has a model ℳ with $⟨ D^{ℳ}, ≺^{ℳ} ⟩$ a well-ordering of type ≥ γ, then ϕ has a model ℳ ’ where $⟨ D^{ℳ^{'}}, ≺^{ℳ^{'}} ⟩$ is non-well-ordered. One of the interesting properties of this number is that the Hanf number of $L_{λ ⁺ ω}$ is exactly $ℶ_{δ (λ)}$ . It was proved in [BK71] that if ℵ₀ < λ < κ $a r e r e g u l a r c a r d i n a l n u m b e r s, t h e n t h e r e i s a f o r c i n g e x t e n s i o n, p r e s e r v i n g c o f i n a l i t i e s, s u c h t h a t i n t h e e x t e n s i o n$ 2λ = κ $a n d δ (λ) < λ ⁺ ⁺ . W e i m p r o v e t h i s r e s u l t b y p r o v i n g t h e f o l l o w i n g : S u p p o s e ℵ ₀ < λ < θ \leq κ a r e c a r d i n a l n u m b e r s s u c h t h a t$ ∙ $λ^{< λ} = λ$ ; ∙ cf(θ) ≥ λ⁺ and $μ^{λ} < θ$ whenever μ < θ; ∙ $κ^{λ} = κ$ . Then there...

Definable orthogonality classes in accessible categories are small

Joan Bagaria, Carles Casacuberta, A. R. D. Mathias, Jiří Rosický (2015)

Journal of the European Mathematical Society

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We lower substantially the strength of the assumptions needed for the validity of certain results in category theory and homotopy theory which were known to follow from Vopěnka’s principle. We prove that the necessary large-cardinal hypotheses depend on the complexity of the formulas defining the given classes, in the sense of the Lévy hierarchy. For example, the statement that, for a class $𝒮$ of morphisms in a locally presentable category $𝒞$ of structures, the orthogonal class of objects...

Explicit construction of normal lattice configurations

Mordechay B. Levin, Meir Smorodinsky (2005)

Colloquium Mathematicae

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We extend Champernowne’s construction of normal numbers to base b to the $ℤ^{d}$ case and obtain an explicit construction of a generic point of the $ℤ^{d}$ shift transformation of the set ${0, 1, . . ., b - 1}^{ℤ^{d}}$ .

Existence and upper semicontinuity of uniform attractors in $H ¹ (ℝ^{N})$ for nonautonomous nonclassical diffusion equations

Cung The Anh, Nguyen Duong Toan (2014)

Annales Polonici Mathematici

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We prove the existence of uniform attractors $_{ε}$ in the space $H ¹ (ℝ^{N})$ for the nonautonomous nonclassical diffusion equation $u_{t} - ε Δ u_{t} - Δ u + f (x, u) + λ u = g (x, t)$ , ε ∈ [0,1]. The upper semicontinuity of the uniform attractors ${_{ε}}_{ε \in [0, 1]}$ at ε = 0 is also studied.

$ℓ_{\infty}$ -sums and the Banach space $ℓ_{\infty} / c ₀$

Christina Brech, Piotr Koszmider (2014)

Fundamenta Mathematicae

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This paper is concerned with the isomorphic structure of the Banach space $ℓ_{\infty} / c ₀$ and how it depends on combinatorial tools whose existence is consistent with but not provable from the usual axioms of ZFC. Our main global result is that it is consistent that $ℓ_{\infty} / c ₀$ does not have an orthogonal $ℓ_{\infty}$ -decomposition, that is, it is not of the form $ℓ_{\infty} (X)$ for any Banach space X. The main local result is that it is consistent that $ℓ_{\infty} (c ₀ ())$ does not embed isomorphically into $ℓ_{\infty} / c ₀$ , where is the cardinality of the continuum,...

Spaces with property $(D C (ω_{1}))$

Wei-Feng Xuan, Wei-Xue Shi (2017)

Commentationes Mathematicae Universitatis Carolinae

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We prove that if $X$ is a first countable space with property $(D C (ω_{1}))$ and with a $G_{δ}$ -diagonal then the cardinality of $X$ is at most $𝔠$ . We also show that if $X$ is a first countable, DCCC, normal space then the extent of $X$ is at most $𝔠$ .

A new Lindelöf space with points $G_{δ}$

Alan S. Dow (2015)

Commentationes Mathematicae Universitatis Carolinae

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We prove that $♦^{*}$ implies there is a zero-dimensional Hausdorff Lindelöf space of cardinality $2^{ℵ_{1}}$ which has points $G_{δ}$ . In addition, this space has the property that it need not be Lindelöf after countably closed forcing.

An Isomorphic Classification of $C (2^{} \times [0, α])$ Spaces

Elói Medina Galego (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

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We present an extension of the classical isomorphic classification of the Banach spaces C([0,α]) of all real continuous functions defined on the nondenumerable intervals of ordinals [0,α]. As an application, we establish the isomorphic classification of the Banach spaces $C (2^{} \times [0, α])$ of all real continuous functions defined on the compact spaces $2^{} \times [0, α]$ , the topological product of the Cantor cubes $2^{}$ with smaller than the first sequential cardinal, and intervals of ordinal numbers [0,α]. Consequently,...

Internally club and approachable for larger structures

John Krueger (2008)

Fundamenta Mathematicae

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We generalize the notion of a fat subset of a regular cardinal κ to a fat subset of $P_{κ} (X)$ , where κ ⊆ X. Suppose μ < κ, $μ^{< μ} = μ$ , and κ is supercompact. Then there is a generic extension in which κ = μ⁺⁺, and for all regular λ ≥ μ⁺⁺, there are stationarily many N in ${[H (λ)]}^{μ ⁺}$ which are internally club but not internally approachable.

Extension properties of Stone-Čech coronas and proper absolute extensors

A. Chigogidze (2013)

Fundamenta Mathematicae

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We characterize, in terms of X, the extensional dimension of the Stone-Čech corona βX∖X of a locally compact and Lindelöf space X. The non-Lindelöf case is also settled in terms of extending proper maps with values in $I^{τ} ∖ L$ , where L is a finite complex. Further, for a finite complex L, an uncountable cardinal τ and a $Z_{τ}$ -set X in the Tikhonov cube $I^{τ}$ we find a necessary and sufficient condition, in terms of $I^{τ} ∖ X$ , for X to be in the class AE([L]). We also introduce a concept of a proper absolute...

On hereditary normality of $ω^{*}$ , Kunen points and character $ω_{1}$

Sergei Logunov (2021)

Commentationes Mathematicae Universitatis Carolinae

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We show that $ω^{*} ∖ {p}$ is not normal, if $p$ is a limit point of some countable subset of $ω^{*}$ , consisting of points of character $ω_{1}$ . Moreover, such a point $p$ is a Kunen point and a super Kunen point.

On non-normality points, Tychonoff products and Suslin number

Sergei Logunov (2022)

Commentationes Mathematicae Universitatis Carolinae

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Let a space $X$ be Tychonoff product $\prod_{α < τ} X_{α}$ of $τ$ -many Tychonoff nonsingle point spaces $X_{α}$ . Let Suslin number of $X$ be strictly less than the cofinality of $τ$ . Then we show that every point of remainder is a non-normality point of its Čech–Stone compactification $β X$ . In particular, this is true if $X$ is either $R^{τ}$ or $ω^{τ}$ and a cardinal $τ$ is infinite and not countably cofinal.

A Dichotomy Principle for Universal Series

V. Farmaki, V. Nestoridis (2008)

Bulletin of the Polish Academy of Sciences. Mathematics

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Applying results of the infinitary Ramsey theory, namely the dichotomy principle of Galvin-Prikry, we show that for every sequence ${(α_{j})}_{j = 1}^{\infty}$ of scalars, there exists a subsequence ${(α_{k_{j}})}_{j = 1}^{\infty}$ such that either every subsequence of ${(α_{k_{j}})}_{j = 1}^{\infty}$ defines a universal series, or no subsequence of ${(α_{k_{j}})}_{j = 1}^{\infty}$ defines a universal series. In particular examples we decide which of the two cases holds.

On the nontrivial solvability of systems of homogeneous linear equations over $ℤ$ in ZFC

Jan Šaroch (2020)

Commentationes Mathematicae Universitatis Carolinae

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Motivated by the paper by H. Herrlich, E. Tachtsis (2017) we investigate in ZFC the following compactness question: for which uncountable cardinals $κ$ , an arbitrary nonempty system $S$ of homogeneous $ℤ$ -linear equations is nontrivially solvable in $ℤ$ provided that each of its subsystems of cardinality less than $κ$ is nontrivially solvable in $ℤ$ ?

Displaying similar documents to “Some remarks on universality properties of ℓ ∞ / c ₀ ”

Displaying similar documents to “Some remarks on universality properties of $ℓ_{\infty} / c ₀$ ”