On ultrapowers of Banach spaces of type \mathscrL<sub>∞</sub>

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

doi:10.4064/fm222-3-1

Cited by 12 publications

(4 citation statements)

References 28 publications

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“…In fact, since c 0 is complemented in F (c 0 ), by Godefroy-Kalton's theorem [12] it would be isomorphic to a complemented subspace of X U . However that is impossible by Proposition 3.3 in [4].…”

Section: Some Remarks On the Cotype Of Lipschitz-free Spacesmentioning

confidence: 95%

Lipschitz-free spaces, ultraproducts, and finite representability of metric spaces

García-Lirola¹,

Grelier²

2022

Preprint

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We study several properties and applications of the ultrapower M U of a metric space M . We prove that the Lipschitz-free space F (M U ) is finitely representable in F (M ). We also characterize the metric spaces that are finitely Lipschitz representable in a Banach space as those that biLipschitz embed into an ultrapower of the Banach space. Thanks to this link, we obtain that if M is finitely Lipschitz representable in a Banach space X, then F (M ) is finitely representable in F (X). We apply these results to the study of cotype in Lipschitz-free spaces and the stability of Lipschitz-free spaces and spaces of Lipschitz functions under ultraproducts.

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Section: Some Remarks On the Cotype Of Lipschitz-free Spacesmentioning

confidence: 95%

Lipschitz-free spaces, ultraproducts, and finite representability of metric spaces

García-Lirola¹,

Grelier²

2022

Preprint

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“…It can be proved (see [11,Theorem 6.1.8] or [12,Theorem 13.9]) that if U is countably incomplete and ℵ-good, then every family of less than ℵ internal subsets of S i U having the finite intersection property has nonempty intersection. Since |Γ × N| < ℵ and U is ℵ-good, there is x ∈ X i U in the nonempty intersection A combination of [4] and [3], see also [5], shows that there exist universally 1-separably injective spaces not isomorphic to any C(K) space. A higher cardinal generalization is as follows.…”

Section: Ultraproductsmentioning

confidence: 97%

$\aleph$-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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“…It had been already observed in [AvCSCaGoMo,Proposition 4.13] that any non-trivial ultrapower of G is ultrahomogeneous. We shall now see that Fraïssé Banach spaces are locally determined.…”

Section: Consequentlymentioning

confidence: 99%

Amalgamation and Ramsey properties of $L_p$ spaces

Ferenczi,

Lopez-Abad,

Mbombo

et al. 2019

Preprint

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We study the dynamics of the group of isometries of Lp-spaces. In particular, we study the canonical actions of these groups on the space of δ-isometric embeddings of finite dimensional subspaces of Lp(0, 1) into itself, and we show that for p = 4, 6, 8, . . . they are ε-transitive provided that δ is small enough. We achieve this by extending the classical equimeasurability principle of Plotkin and Rudin. We define the central notion of a Fraïssé Banach space which underlies these results and of which the known separable examples are the spaces Lp(0, 1), p = 4, 6, 8, . . . and the Gurarij space. We also give a proof of the Ramsey property of the classes { n p }n, p = 2, ∞, viewing it as a multidimensional Borsuk-Ulam statement. We relate this to an arithmetic version of the Dual Ramsey Theorem of Graham and Rothschild as well as to the notion of a spreading vector of Matoušek and Rödl. Finally, we give a version of the Kechris-Pestov-Todorcevic correspondence that links the dynamics of the group of isometries of an approximately ultrahomogeneous space X with a Ramsey property of the collection of finite dimensional subspaces of X.

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On ultrapowers of Banach spaces of type \mathscrL_∞

Cited by 12 publications

References 28 publications

Lipschitz-free spaces, ultraproducts, and finite representability of metric spaces

Lipschitz-free spaces, ultraproducts, and finite representability of metric spaces

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

Amalgamation and Ramsey properties of $L_p$ spaces

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On ultrapowers of Banach spaces of type \mathscrL∞

Cited by 12 publications

References 28 publications

Lipschitz-free spaces, ultraproducts, and finite representability of metric spaces

Lipschitz-free spaces, ultraproducts, and finite representability of metric spaces

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

Amalgamation and Ramsey properties of $L_p$ spaces

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Product

Resources

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On ultrapowers of Banach spaces of type \mathscrL_∞