On separably injective Banach spaces

Avilés, Antonio; Sánchez, Félix Cabello; Castillo, Jesús M. F.; González, Manuel; Moreno, Yolanda

doi:10.1016/j.aim.2012.10.013

Cited by 51 publications

(141 citation statements)

References 32 publications

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“…In this section we extend some results proved in [4] for separably injective Banach spaces. Recall that a Banach space E is separably injective (ℵ 1 -injective according to Definition 0.1) when E-valued operators extends to separable super-spaces, and that E is universally separably injective (universally ℵ 1 -injective) when E-valued operators extend from separable subspaces.…”

Section: ℵ-Injective Banach Spacesmentioning

confidence: 88%

“…The push-out and pull-back constructions. A thorough description of the pull-back and push-out constructions in Banach spaces can be seen in [4,3,9]. Everything we need to know for this paper is that given an exact sequence (1) and an operator t : Y → B there is a commutative diagram…”

Section: Exact Sequencesmentioning

confidence: 99%

“…Section 1 is preliminary; it contains some definitions together with the minimal background on exact sequences of Banach spaces one needs to read the paper. In Section 2 we extend a variety of results in [4] about (universal) separably injective Banach spaces to higher cardinals. However, we found no reasonable generalization for a considerable portion of the results proved in [4] for ℵ 1 and so the resulting picture is rather incomplete.…”

Section: Introductionmentioning

confidence: 99%

“…In Section 2 we extend a variety of results in [4] about (universal) separably injective Banach spaces to higher cardinals. However, we found no reasonable generalization for a considerable portion of the results proved in [4] for ℵ 1 and so the resulting picture is rather incomplete. In contrast to Section 2, which deals mainly with "isomorphic" properties, the ensuing Section 3 is of "isometric nature" and studies some special properties of (1, ℵ)-injective spaces and their universal relatives.…”

Section: Introductionmentioning

confidence: 99%

“…It is worth noticing that Zippin proved in the late seventies that every infinite dimensional separable and separably injective Banach space has to be isomorphic to c 0 , the space of all null sequences with the sup norm, and so even in the case ℵ = ℵ 1 one is mainly concerned with nonseparable Banach spaces. We refer the reader to [39,4] for an account and further references.…”

Section: Introductionmentioning

confidence: 99%

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$\aleph$-injective Banach spaces and $\aleph$-projective compacta

Avilés¹,

Sánchez²,

Castillo³

et al. 2015

Rev. Mat. Iberoam.

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Abstract. A Banach space E is said to be injective if for every Banach space X and every subspace Y of X every operator t : Y → E has an extension T : X → E. We say that E is ℵ-injective (respectively, universally ℵ-injective) if the preceding condition holds for Banach spaces X (respectively Y ) with density less than a given uncountable cardinal ℵ. We perform a study of ℵ-injective and universally ℵ-injective Banach spaces which extends the basic case where ℵ = ℵ 1 is the first uncountable cardinal. When dealing with the corresponding "isometric" properties we arrive to our main examples: ultraproducts and spaces of type C(K). We prove that ultraproducts built on countably incomplete ℵ-good ultrafilters are (1, ℵ)-injective as long as they are Lindenstrauss spaces. We characterize (1, ℵ)-injective C(K) spaces as those in which the compact K is an F ℵ -space (disjoint open subsets which are the union of less than ℵ many closed sets have disjoint closures) and we uncover some projectiveness properties of F ℵ -spaces.

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Section: ℵ-Injective Banach Spacesmentioning

confidence: 88%

Section: Exact Sequencesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

$\aleph$-injective Banach spaces and $\aleph$-projective compacta

Avilés¹,

Sánchez²,

Castillo³

et al. 2015

Rev. Mat. Iberoam.

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Universal mappings for certain classes of operators and polynomials between Banach spaces

Cilia

Gutiérrez

2017

Mathematische Nachrichten

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A well-known result of J. Lindenstrauss and A. Pełczyński (1968) gives the existence of a universal non-weakly compact operator between Banach spaces. We show the existence of universal non-Rosenthal, non-limited, and non-Grothendieck operators.We also prove that there does not exist a universal non-Dunford-Pettis operator, but there is a universal class of non-Dunford-Pettis operators. Moreover, we show that, for several classes of polynomials between Banach spaces, including the non-weakly compact polynomials, there does not exist a universal polynomial. K E Y W O R D SIdeals of homogeneous polynomials, surjective operator ideals, universal operator, universal polynomial M S C ( 2 0 1 0 ) Johnson [27] showed in 1971 that the formal identity operator 1 → ∞ is universal for the class of non-compact operators. These results are useful in order to prove that a given operator is non-weakly compact (respectively, non-compact). In 1997, M. Girardi and W. B. Johnson [21] proved that there does not exist a universal non-completely continuous operator, but there is a class  of universal non-completely continuous operators, that is, for every non-completely continuous operator , there is some member of  that factors through .Here we prove the existence of universal operators for the classes of non-Rosenthal, non-Grothendieck, and non-limited operators and the existence of a universal class of non-Dunford-Pettis operators (see below the definitions of all such classes

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