Remarks on the complementability of spaces of Bochner integrable functions in spaces of vector measures

Giovanni Emmanuele

Displaying similar documents to “Remarks on the complementability of spaces of Bochner integrable functions in spaces of vector measures”

On ultrapowers of Banach spaces of type $ℒ_{\infty}$

Antonio Avilés, Félix Cabello Sánchez, Jesús M. F. Castillo, Manuel González, Yolanda Moreno (2013)

Fundamenta Mathematicae

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We prove that no ultraproduct of Banach spaces via a countably incomplete ultrafilter can contain c₀ complemented. This shows that a "result" widely used in the theory of ultraproducts is wrong. We then amend a number of results whose proofs have been infected by that statement. In particular we provide proofs for the following statements: (i) All M-spaces, in particular all C(K)-spaces, have ultrapowers isomorphic to ultrapowers of c₀, as also do all their complemented subspaces isomorphic...

Banach spaces widely complemented in each other

Elói Medina Galego (2013)

Colloquium Mathematicae

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Suppose that X and Y are Banach spaces that embed complementably into each other. Are X and Y necessarily isomorphic? In this generality, the answer is no, as proved by W. T. Gowers in 1996. However, if X contains a complemented copy of its square X², then X is isomorphic to Y whenever there exists p ∈ ℕ such that $X^{p}$ can be decomposed into a direct sum of $X^{p - 1}$ and Y. Motivated by this fact, we introduce the concept of (p,q,r) widely complemented subspaces in Banach spaces, where p,q and...

On copies of $c_{0}$ in the bounded linear operator space

Juan Carlos Ferrando, J. M. Amigó (2000)

Czechoslovak Mathematical Journal

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In this note we study some properties concerning certain copies of the classic Banach space $c_{0}$ in the Banach space $ℒ (X, Y)$ of all bounded linear operators between a normed space $X$ and a Banach space $Y$ equipped with the norm of the uniform convergence of operators.

The structure of Lindenstrauss-Pełczyński spaces

Jesús M. F. Castillo, Yolanda Moreno, Jesús Suárez (2009)

Studia Mathematica

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Lindenstrauss-Pełczyński (for short ℒ) spaces were introduced by these authors [Studia Math. 174 (2006)] as those Banach spaces X such that every operator from a subspace of c₀ into X can be extended to the whole c₀. Here we obtain the following structure theorem: a separable Banach space X is an ℒ-space if and only if every subspace of c₀ is placed in X in a unique position, up to automorphisms of X. This, in combination with a result of Kalton [New York J. Math. 13 (2007)], provides...

On complemented copies of c₀(ω₁) in C(Kⁿ) spaces

Leandro Candido, Piotr Koszmider (2016)

Studia Mathematica

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Given a compact Hausdorff space K we consider the Banach space of real continuous functions C(Kⁿ) or equivalently the n-fold injective tensor product $\otimes ̂_{ε}^{n} C (K)$ or the Banach space of vector valued continuous functions C(K,C(K,C(K...,C(K)...). We address the question of the existence of complemented copies of c₀(ω₁) in $\otimes ̂_{ε}^{n} C (K)$ under the hypothesis that C(K) contains such a copy. This is related to the results of E. Saab and P. Saab that $X \otimes ̂_{ε} Y$ contains a complemented copy of c₀ if one of the infinite-dimensional...

Isomorphic and isometric copies of $ℓ_{\infty} (Γ)$ in duals of Banach spaces and Banach lattices

Marek Wójtowicz (2006)

Commentationes Mathematicae Universitatis Carolinae

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Let $X$ and $E$ be a Banach space and a real Banach lattice, respectively, and let $Γ$ denote an infinite set. We give concise proofs of the following results: (1) The dual space $X^{*}$ contains an isometric copy of $c_{0}$ iff $X^{*}$ contains an isometric copy of $ℓ_{\infty}$ , and (2) $E^{*}$ contains a lattice-isometric copy of $c_{0} (Γ)$ iff $E^{*}$ contains a lattice-isometric copy of $ℓ_{\infty} (Γ)$ .

Projections from $L (X, Y)$ onto $K (X, Y)$

Kamil John (2000)

Commentationes Mathematicae Universitatis Carolinae

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Generalization of certain results in [Sap] and simplification of the proofs are given. We observe e.g.: Let $X$ and $Y$ be Banach spaces such that $X$ is weakly compactly generated Asplund space and $X^{*}$ has the approximation property (respectively $Y$ is weakly compactly generated Asplund space and $Y^{*}$ has the approximation property). Suppose that $L (X, Y) \neq K (X, Y)$ and let $1 < λ < 2$ . Then $X$ (respectively $Y$ ) can be equivalently renormed so that any projection $P$ of $L (X, Y)$ onto $K (X, Y)$ has the sup-norm greater or equal to $λ$ . ...

The Dual of a Non-reflexive L-embedded Banach Space Contains $l^{\infty}$ Isometrically

Hermann Pfitzner (2010)

Bulletin of the Polish Academy of Sciences. Mathematics

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A Banach space is said to be L-embedded if it is complemented in its bidual in such a way that the norm between the two complementary subspaces is additive. We prove that the dual of a non-reflexive L-embedded Banach space contains $l^{\infty}$ isometrically.

The Maurey extension property for Banach spaces with the Gordon-Lewis property and related structures

P. G. Casazza, N. J. Nielsen (2003)

Studia Mathematica

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The main result of this paper states that if a Banach space X has the property that every bounded operator from an arbitrary subspace of X into an arbitrary Banach space of cotype 2 extends to a bounded operator on X, then every operator from X to an L₁-space factors through a Hilbert space, or equivalently $B (ℓ_{\infty}, X *) = Π ₂ (ℓ_{\infty}, X *)$ . If in addition X has the Gaussian average property, then it is of type 2. This implies that the same conclusion holds if X has the Gordon-Lewis property (in particular X could...

The problem of complementability for some spaces of vector measures of bounded variation with values in Banach spaces containing copies of $c_{0}$

L. Drewnowski, G. Emmanuele (1993)

Studia Mathematica

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Let (S, ∑, m) be any atomless finite measure space, and X any Banach space containing a copy of $c_{0}$ . Then the Bochner space $L^{1} (m; X)$ is uncomplemented in ccabv(∑,m;X), the Banach space of all m-continuous vector measures that are of bounded variation and have a relatively compact range; and ccabv(∑,m;X) is uncomplemented in cabv(∑,m;X). It is conjectured that this should generalize to all Banach spaces X without the Radon-Nikodym property.