The isomorphism class of C0 is not Borel

Kurka, Ondřej

doi:10.1007/s11856-019-1851-0

Cited by 4 publications

(5 citation statements)

References 20 publications

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“…There is an active ongoing research whether for a particular Banach space its isomorphism class is Borel or not (see, e.g. [26], [36], [22], [27]), while it is known that isometry classes of separable Banach spaces are always Borel (note that the linear isometry relation is Borel bireducible with an orbit equivalence relation [44], and orbit equivalence relations have Borel equivalence classes [34,Theorem 15.14]). Having a topology at our disposal, we compute complexities of isometry classes of several classical Banach spaces.…”

Section: Theorem Amentioning

confidence: 99%

“…This concerns, for example, reflexive spaces, spaces with separable dual, spaces containing 1 , spaces with the Radon-Nikodým property, spaces isomorphic to L p [0,1] for p ∈ (1,2) ∪ (2,∞), or spaces isomorphic to c 0 . We refer to [7, page 130 and Corollary 3.3] and [36,Theorem 1.1] for papers which contain the corresponding results and to the monograph [17] and the survey [26] for some more information. Thus, for example, in combination with Corollary 5.4, we see that none of those classes might be characterized as a class into which a given locally finite metric space bi-Lipchitz embeds.…”

Section: Superreflexive Spacesmentioning

confidence: 99%

See 1 more Smart Citation

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

Cúth,

Doležal,

Doucha

et al. 2023

J. Inst. Math. Jussieu

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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 $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.

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Section: Theorem Amentioning

confidence: 99%

Section: Superreflexive Spacesmentioning

confidence: 99%

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

Cúth,

Doležal,

Doucha

et al. 2023

J. Inst. Math. Jussieu

Self Cite

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“…reflexive spaces, spaces with separable dual, spaces containing ℓ 1 , spaces with the Radon-Nikodým property, spaces isomorphic to L p [0, 1] for p ∈ (1, 2) ∪ (2, ∞), or spaces isomorphic to c 0 . We refer to [7, page 130 and Corollary 3.3] and [36,Theorem 1.1] for papers which contain the corresponding results and to the monograph [17] and the survey [26] for some more information. Thus, e.g.…”

Section: 2mentioning

confidence: 99%

“…There is an active ongoing research whether for a particular Banach space its isomorphism class is Borel or not (see e.g. [26], [36], [22], [27]) while it is known that isometry classes of separable Banach spaces are always Borel (note that the linear isometry relation is Borel bi-reducible with an orbit equivalence relation [44] and orbit equivalence relations have Borel equivalence classes [34,Theorem 15.14]). Having a topology at our disposal we compute complexities of isometry classes of several classical Banach spaces.…”

Section: Introductionmentioning

confidence: 99%

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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“…Moreover, one of the active and ongoing research streams is to find out whether for a particular Banach space its isomorphism class is Borel or not (see, e.g., [25, 18] or the survey [17] and references therein): spaces with Borel isomorphism classes are rare and can be considered simply definable (up to isomorphism). It is then desirable, for spaces whose classes are Borel, to have a finer description of how simply definable they are (see, e.g., [18, Problem 3]).…”

Section: Introductionmentioning

confidence: 99%

Polish spaces of Banach spaces

Cúth

Doležal

Doucha

et al. 2022

Forum of Mathematics, Sigma

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We present and thoroughly study natural Polish spaces of separable Banach spaces. These spaces are defined as spaces of norms, respectively pseudonorms, on the countable infinite-dimensional rational vector space. We provide an exhaustive comparison of these spaces with admissible topologies recently introduced by Godefroy and Saint-Raymond and show that Borel complexities differ little with respect to these two topological approaches. We investigate generic properties in these spaces and compare them with those in admissible topologies, confirming the suspicion of Godefroy and Saint-Raymond that they depend on the choice of the admissible topology.

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The isomorphism class of C0 is not Borel

Cited by 4 publications

References 20 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

Polish spaces of Banach spaces

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