On complementably universal Banach spaces

Kadec, M. I.

doi:10.4064/sm-40-1-85-89

Cited by 23 publications

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“…2 Theorem 2.1 suggests the following problem: What can one say about a separable Banach space which is complementably universal for the collection of all subspaces of c 0 which have the BAP? By the results of Kadec [9] and Pełczyński [12] mentioned in the introduction, such spaces do exist. The most natural question is whether a subspace of c 0 can have this universal property.…”

Section: Resultsmentioning

confidence: 93%

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Complementably universal Banach spaces, II

Johnson

Szankowski

2009

Journal of Functional Analysis

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Section: Resultsmentioning

confidence: 93%

“…Kadec [9] subsequently constructed a separable Banach space with the bounded approximation property (BAP) which is complementably universal for the class of all separable Banach spaces which have the BAP. Actually, the spaces constructed by Kadec and Pełczyński are isomorphic (see [6] and [13]).…”

Section: Introductionmentioning

confidence: 99%

Complementably universal Banach spaces, II

Johnson

Szankowski

2009

Journal of Functional Analysis

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“…For example, given a family F of operators between Banach spaces, it is natural to try to find a single (usually separable) Banach space Z such that all the operators in F factor through Z. If F is the collection of all operators between separable Banach spaces that have the BAP, there is such a separable Z; namely, the separable universal basis space of Pełczyński [1969], Pełczyński [1971], Kadec [1971]. This space, as well as smaller (even reflexive) spaces W. B.…”

Section: Approximation Propertiesmentioning

confidence: 99%

Some 20+ Year Old Problems About Banach Spaces and Operators on Them

Johnson¹

2019

Proceedings of the International Congress of Mathematicians (ICM 2018)

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In the last few years numerous 20+ year old problems in the geometry of Banach spaces were solved. Some are described herein. The diameter of the isomorphism class of a Banach space Given a Banach space X, define D(X) = supfd (X 1 ; X 2 ) : X 1 ; X 2 are isomorphic to Xg: J. J. Schäffer considered the following question to be well-known when he included it in his 1976 book Schäffer [1976]:Is D(X) = 1 for all infinite dimensional X? My first PhD student, E. Odell (who died much too young) and I gave an affirmative answer for separable X W. B. Johnson and E. Odell [2005]. A new definition helped: Call a Banach space X K-elastic provided every isomorph of X K-embeds into X. Call X elastic if X is K-elastic for some K < 1. Ted and I proved the following, which easily implies that D(X) = 1 if X is separable and infinite dimensional.The only "obvious" example of a separable elastic space is C [0; 1]. It is 1-elastic because Mazur proved that every separable Banach space is isometrically isomorphic to a subspace of C [0; 1]. Odell and I suspected that isomorphs of C [0; 1] are the only elastic separable spaces and remarked that our proof of Theorem 2.1 could be streamlined a lot if this is true. We could not prove this but were able to use Bourgain's`1 index theory Bourgain [1980] to prove that a separable elastic space contains a subspace that is isomorphic to c 0 and used that information in the proof of Theorem 2.1. Ten years later my third PhD student, D. Alspach, and B. Sari created a new index that they used to verify that our suspicion was correct. Their proof is rather complicated, but even more recently Beanland and Causey [n.d.] simplified the proof somewhat by using more descriptive set theory. It looks likely that the Alspach-Sari index will be used more down the road. Schäffer's problem remains open for non separable spaces. In some models of set theory (GCH) there are spaces of every density character that are 1-elastic by virtue of being universal, but some tools that were used in the separable setting are not available when the spaces are non separable. Godefroy [2010] proved that under Martin's Maximum Axiom Schäffer's problem has an affirmative answer for subspaces of`1. CommutatorsThe commutator of two elements A and B in a Banach algebra is given by [A; B] = AB BA:A natural problem that arises in the study of derivations on a Banach algebra A is to classify the commutators in the algebra. Probably the most natural non commutative

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“…Recall that a Banach space X is complementably universal for a class M of Banach space provided that every space in M is isomorphic to a complemented subspace of X. Kadec [12] constructed a separable Banach space with the BAP that is complementably universal for all separable Banach spaces that have the BAP, while the authors [10] proved that there is no separable Banach space that is complementably universal for all separable Banach spaces that have the AP. Timur Oikhberg asked the authors whether there is a separable infinite dimensional Banach space not isomorphic to 2 that is complementably universal for all subspaces of itself.…”

Section: More Applicationsmentioning

confidence: 99%

Hereditary approximation property

Johnson

Szankowski

2012

Ann. Math.

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If X is a Banach space such that the isomorphism constant to n 2 from n-dimensional subspaces grows sufficiently slowly as n → ∞, then X has the approximation property. A consequence of this is that there is a Banach space X with a symmetric basis but not isomorphic to 2 so that all subspaces of X have the approximation property. This answers a problem raised in 1980. An application of the main result is that there is a separable Banach space X that is not isomorphic to a Hilbert space, yet every subspace of X is isomorphic to a complemented subspace of X. This contrasts with the classical result of Lindenstrauss and Tzafriri that a Banach space in which every closed subspace is complemented must be isomorphic to a Hilbert space.

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On complementably universal Banach spaces

Cited by 23 publications

References 0 publications

Complementably universal Banach spaces, II

Complementably universal Banach spaces, II

Some 20+ Year Old Problems About Banach Spaces and Operators on Them

Hereditary approximation property

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