Uniformly factoring weakly compact operators and parametrised dualisation

Antunes, Leandro; Beanland, Kevin; Braga, Bruno M.

doi:10.1017/fms.2020.68

Cited by 2 publications

(3 citation statements)

References 21 publications

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“…(2) For p ∈ [1, 2) ∪ (2, ∞), the isometry class of ℓ p is an F σδ -complete set (see Theorem 4.1). (3) The isometry class of c 0 is an F σδ -complete set (see Theorem 4.1). (4) The isometry class of the Gurariȋ space is a G δ -complete set (see Corollary 3.2).…”

Section: Theorem Bmentioning

confidence: 99%

“…Descriptive set theoretic approach to Banach spaces has proved to be a powerful tool in solving many problems in Banach space theory; for a wide selection of references ranging from the earliest ones of Bourgain to the most recent ones see e.g. [8,4,5,3,12]. Traditionally, and as defined explicitly for the first time in the seminal papers of Bossard ([6], [7]), one considers the standard Borel space of all separable Banach spaces, which can be defined as an appropriate Borel subspace of the Effros-Borel space of all closed subspaces of some isometrically universal separable Banach space.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Polish spaces of Banach spaces. Complexity of isometry and isomorphism classes

Cúth¹,

Doležal²,

Doucha³

et al. 2022

Preprint

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We study the complexities of isometry and isomorphism classes of separable Banach spaces in the Polish spaces of Banach spaces recently introduced and investigated by the authors in [14]. We obtain sharp results concerning the most classical separable Banach spaces.We prove that the infinite-dimensional separable Hilbert space is characterized as the unique separable infinite-dimensional Banach space whose isometry class is closed, and also as the unique separable infinite-dimensional Banach space whose isomorphism class is Fσ. For p ∈ [1, 2) ∪ (2, ∞), we show that the isometry classes of Lp[0, 1] and ℓp are G δ -complete sets and F σδ -complete sets, respectively. Then we show that the isometry class of c 0 is an F σδ -complete set.Additionally, we compute the complexities of many other natural classes of separable Banach spaces; for instance, the class of separable L p,λ+ -spaces, for p, λ ≥ 1, is shown to be a G δ -set, the class of superreflexive spaces is shown to be an F σδ -set, and the class of spaces with local Π-basis structure is shown to be a Σ 0 6 -set. The paper is concluded with many open problems and suggestions for a future research.

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Section: Theorem Bmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Polish spaces of Banach spaces. Complexity of isometry and isomorphism classes

Cúth¹,

Doležal²,

Doucha³

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

“…Descriptive set theoretic approach to Banach spaces has proved to be a powerful tool in solving many problems in Banach space theory; for a wide selection of references ranging from the earliest ones of Bourgain to the most recent ones see e.g. [3,4,5,8,12]. Traditionally, and as defined explicitly for the first time in the seminal papers of Bossard ([6, 7]), one considers the standard Borel space of all separable Banach spaces, which can be defined as an appropriate Borel subspace of the Effros-Borel space of all closed subspaces of some isometrically universal separable Banach space.…”

Section: Introductionmentioning

confidence: 99%

Polish Spaces of Banach Spaces: Complexity of Isometry and Isomorphism Classes

Cúth,

Doležal,

Doucha

et al. 2023

J. Inst. Math. Jussieu

View full text Add to dashboard Cite

We study the complexities of isometry and isomorphism classes of separable Banach spaces in the Polish spaces of Banach spaces, recently introduced and investigated by the authors in [14]. We obtain sharp results concerning the most classical separable Banach spaces. We prove that the infinite-dimensional separable Hilbert space is characterized as the unique separable infinite-dimensional Banach space whose isometry class is closed, and also as the unique separable infinite-dimensional Banach space whose isomorphism class is $F_\sigma $ . For $p\in \left [1,2\right )\cup \left (2,\infty \right )$ , we show that the isometry classes of $L_p[0,1]$ and $\ell _p$ are $G_\delta $ -complete sets and $F_{\sigma \delta }$ -complete sets, respectively. Then we show that the isometry class of $c_0$ is an $F_{\sigma \delta }$ -complete set. Additionally, we compute the complexities of many other natural classes of separable Banach spaces; for instance, the class of separable $\mathcal {L}_{p,\lambda +}$ -spaces, for $p,\lambda \geq 1$ , is shown to be a $G_\delta $ -set, the class of superreflexive spaces is shown to be an $F_{\sigma \delta }$ -set, and the class of spaces with local $\Pi $ -basis structure is shown to be a $\boldsymbol {\Sigma }^0_6$ -set. The paper is concluded with many open problems and suggestions for a future research.

show abstract

Uniformly factoring weakly compact operators and parametrised dualisation

Cited by 2 publications

References 21 publications

Polish spaces of Banach spaces. Complexity of isometry and isomorphism classes

Polish spaces of Banach spaces. Complexity of isometry and isomorphism classes

Polish Spaces of Banach Spaces: Complexity of Isometry and Isomorphism Classes

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