Some equivalence relations which are borel reducible to isomorphism between separable banach spaces

Ferenczi, Valentin; Galego, Elói Medina

doi:10.1007/bf02771976

Cited by 14 publications

(30 citation statements)

References 31 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This result is the culmination of a series of successive lower estimates of the complexity by Bossard [4], Rosendal [19], and Ferenczi-Galego [8]. We also consider the corresponding quasiorders of embeddability etc.…”

Section: Theorem 5 the Relations Of Isomorphism And Lipschitz Isomormentioning

confidence: 99%

The complexity of classifying separable Banach spaces up to isomorphism

Ferenczi¹,

Louveau²,

Rosendal³

2009

Journal of the London Mathematical Society

Self Cite

View full text Add to dashboard Cite

Abstract. It is proved that the relation of isomorphism between separable Banach spaces is a complete analytic equivalence relation, i.e., that any analytic equivalence relation Borel reduces to it. Thus, separable Banach spaces up to isomorphism provide complete invariants for a great number of mathematical structures up to their corresponding notion of isomorphism. The same is shown to hold for (1) complete separable metric spaces up to uniform homeomorphism, (2) separable Banach spaces up to Lipschitz isomorphism, and (3) up to (complemented) biembeddability, (4) Polish groups up to topological isomorphism, and (5) Schauder bases up to permutative equivalence. Some of the constructions rely on methods recently developed by S. Argyros and P. Dodos.

show abstract

Section: Theorem 5 the Relations Of Isomorphism And Lipschitz Isomormentioning

confidence: 99%

The complexity of classifying separable Banach spaces up to isomorphism

Ferenczi¹,

Louveau²,

Rosendal³

2009

Journal of the London Mathematical Society

Self Cite

View full text Add to dashboard Cite

show abstract

“…Then using Gowers's dichotomy we can find a subspace Y with an unconditional basis. Now, a recent result due to Ferenczi and Galego [7] says that E 0 Borel reduces to the isomorphism relation between subspaces of c 0 and 1 . So by James's characterisation of reflexivity, this basis must span a reflexive space, and by applying Proposition 17, we have our result.…”

Section: Proof Since [N]mentioning

confidence: 99%

“…A continuum? Even for some of the classical spaces this question is still open, though recently progress has been made by Ferenczi and Galego [7].…”

mentioning

confidence: 99%

Incomparable, non-isomorphic and minimal Banach spaces

Rosendal

2004

Fund. Math.

View full text Add to dashboard Cite

Abstract. A Banach space contains either a minimal subspace or a continuum of incomparable subspaces. General structure results for analytic equivalence relations are applied in the context of Banach spaces to show that if E 0 does not reduce to isomorphism of the subspaces of a space, in particular, if the subspaces of the space admit a classification up to isomorphism by real numbers, then any subspace with an unconditional basis is isomorphic to its square and hyperplanes, and the unconditional basis has an isomorphically homogeneous subsequence.

show abstract

“…Concerning those spaces, it is known that c 0 and p , 1 p < 2 [9] are ergodic. By [29], the space T is ergodic, and the proof holds to show that T * is ergodic as well.…”

Section: Introductionmentioning

confidence: 98%

On the number of permutatively inequivalent basic sequences in a Banach space

Ferenczi¹

2006

Journal of Functional Analysis

Self Cite

View full text Add to dashboard Cite

Let X be a Banach space with a Schauder basis (e n ) n∈N . The relation E 0 is Borel reducible to permutative equivalence between normalized block-sequences of (e n ) n∈N or X is c 0 or p saturated for some 1 p < +∞. If (e n ) n∈N is shrinking unconditional then either it is equivalent to the canonical basis of c 0 or p , 1 < p < +∞, or the relation E 0 is Borel reducible to permutative equivalence between sequences of normalized disjoint blocks of X or of X * . If (e n ) n∈N is unconditional, then either X is isomorphic to 2 , or X contains 2 ω subspaces or 2 ω quotients which are spanned by pairwise permutatively inequivalent normalized unconditional bases.

show abstract

Some equivalence relations which are borel reducible to isomorphism between separable banach spaces

Cited by 14 publications

References 31 publications

The complexity of classifying separable Banach spaces up to isomorphism

The complexity of classifying separable Banach spaces up to isomorphism

Incomparable, non-isomorphic and minimal Banach spaces

On the number of permutatively inequivalent basic sequences in a Banach space

Contact Info

Product

Resources

About