On Equivalence Relations Generated by Schauder Bases

Ding, Lei

doi:10.1017/jsl.2017.67

Cited by 4 publications

(4 citation statements)

References 22 publications

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“…Lemma 5.8 of [8] concerning finite dimensional spaces is a special case of the following theorem. Theorem 6.14.…”

Section: Now We Define π : ({1mentioning

confidence: 99%

On equivalence relations induced by Polish groups

Ding¹,

Zheng²

2022

Preprint

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The motivation of this article is to introduce a kind of orbit equivalence relations which can well describe structures and properties of Polish groups from the perspective of Borel reducibility. Given a Polish group G, let E(G) be the right coset equivalence relation G ω /c(G), where c(G) is the group of all convergent sequences in G.Let G be a Polish group.(1) G is a discrete countable group containing at least two elements iffThe notion of α-unbalanced Polish group for α < ω1 is introduced. Let G, H be Polish groups, 0For any Lie group G, denote G0 the connected component of the identity element 1G. Let G and H be two separable TSI Lie groups. If E(G) ≤B E(H), then there exists a continuous locally injection S : G0 → H0. Moreover, if G0, H0 are abelian, S is a group homomorphism. Particularly, for c0, e0, c1, e1 ∈ N, E(R c 0 × T e 0 ) ≤B E(R c 1 × T e 1 ) iff e0 ≤ e1 and c0 + e0 ≤ c1 + e1.

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“…Lemma 5.8 of [8] concerning finite dimensional spaces is a special case of the following theorem. Theorem 6.14.…”

Section: Now We Define π : ({1mentioning

confidence: 99%

On equivalence relations induced by Polish groups

Ding¹,

Zheng²

2022

Preprint

Self Cite

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“…x n converges unconditionally means that the series Definition 2.2 (Ding [6]). For a basic sequence {x n } in Banach space X, we denote coef(X, (x n )) to be the set of all a = (a n ) ∈ R N such that a n x n converges.…”

Section: Preliminariesmentioning

confidence: 99%

“…We define an equivalence relation

E (X, false(x_{n} false))

R^{ω}

a E false(X, (x_{n}) false) b : \Leftrightarrow a - b \in coef false(X, (x_{n}) false)

and call equivalence relations of this form Schauder equivalence relations . This definition was introduced by Ding in .…”

Section: Introductionmentioning

confidence: 99%

On Schauder equivalence relations

2017

Mathematical Logic Qtrly

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Abstract. In this paper, we study Schauder equivalence relations, which are Borel equivalence relations generated by Banach spaces with basic sequences.We prove that the set of equivalence relations generated by basic sequences has boundaries. In addition, we prove that both R N /lp and R N /c 0 are not reducible to the equivalence relation generated by Tsirelson space T with the unit vector basis {tn}. We also shows that Borel equivalence relations generated by α−Tsirelson spaces are mutually incompatible. Based on this argument, we show that any basis of Schauder equivalence relations must be of cardinal 2 ω .

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“…Theorem 1.1 entails in particular a negative answer to a Question 6.3(1) from [Din17]. Let X be a separable Banach space with a Schauder basis (x n ) n∈N .…”

Section: Introductionmentioning

confidence: 99%

Complexity classes of Polishable subgroups

Lupini¹

2022

Preprint

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In this paper we further develop the theory of canonical approximations of Polishable subgroups of Polish groups, building on previous work of Solecki and Farah-Solecki. In particular, we obtain a characterization of such canonical approximations in terms of their Borel complexity class. As an application we provide a complete list of all the possible Borel complexity classes of Polishable subgroups of Polish groups or, equivalently, of the ranges of continuous group homomorphisms between Polish groups. We also provide a complete list of all the possible Borel complexity classes of the ranges of: continuous group homomorphisms between non-Archimedean Polish groups; continuous linear maps between separable Fréchet spaces; continuous linear maps between separable Banach spaces.

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On Equivalence Relations Generated by Schauder Bases

Cited by 4 publications

References 22 publications

On equivalence relations induced by Polish groups

On equivalence relations induced by Polish groups

On Schauder equivalence relations

Complexity classes of Polishable subgroups

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