Quasi-minimality and tightness by range in spaces with unconditional basis

Argyros, Spiros A.; Manoussakis, Antonis; Pelczar-Barwacz, Anna

doi:10.1007/s11856-014-0004-8

Cited by 2 publications

(13 citation statements)

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“…The list of examples of the main classes was completed by the recent work of V. Ferenczi and Th. Schlumprecht [12] and by the recent work of S.A. Argyros and the authors [6]. We recall now the list of classes developed in [10] as stated in [12], mentioning also some already known examples.…”

Section: Introductionmentioning

confidence: 97%

“…We recall now the list of classes developed in [10] as stated in [12], mentioning also some already known examples. (1) HI, tight by range (Gowers space with asymptotically unconditional basis of [14] by [11]), (2) HI, tight, sequentially minimal (version of Gowers-Maurey space by [12]), (3) tight by support (Gowers space with unconditional basis of [13] by [11]), (4) with unconditional basis, tight by range, quasi-minimal (unconditional version of Gowers HI space with an asymptotically unconditional basis of [14] by [6]), (5) with unconditional basis, tight, sequentially minimal (Tsirelson space by [11]), (6) with unconditional basis, minimal (ℓ p , c 0 , dual to Tsirelson space by [9], Schlumprecht space by [2]). …”

Section: Introductionmentioning

confidence: 99%

“…We briefly sketch the proof of local minimality of the space X (4) constructed in [6] and concentrate on the construction of an example on the other edge of the class (4). Namely we prove the following.…”

Section: Introductionmentioning

confidence: 99%

“…As it was mentioned above the example of [6] is an unconditional version of the Gowers HI space with an asymptotically unconditional basis [13], using the standard framework of a space constructed on the basis of a mixed Tsirelson space, defined by a norming set closed under certain operations. The standard operations include taking averages of certain block sequences (as (A n , θ n -operations or (S n , θ n )-operations), projections on the subsets of N or intervals in N, change of signs etc.…”

Section: Introductionmentioning

confidence: 99%

“…In the case of the space X (4) in [6] its norming set is closed under change of signs, certain (A nj , 1 mj )-operations, projections on intervals, and -in order to ensure tightness by range -under the special "Gowers operation" used in [13], i.e. scaled projection on S 1 sets.…”

Section: Introductionmentioning

confidence: 99%

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Types of tightness in spaces with unconditional basis

Manoussakis

Pelczar-Barwacz

2014

Studia Math.

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Abstract. We present a reflexive Banach space with an unconditional basis which is quasi-minimal and tight by range, i.e. of type (4) in Ferenczi-Rosendal list within the framework of Gowers' classification program of Banach spaces, but contrary to the recently constructed space of type (4) also tight with constants, thus essentially extending the list of known examples in Gowers' program. The space is defined on the base on a boundedly modified mixed Tsirelson space with use of a special coding function. IntroductionThe "loose" classification program for Banach spaces was started by W.T. Gowers in the celebrated paper [16]. The goal is to identify classes of Banach spaces which are• hereditary, i.e. if a space belongs to a given class, then any its closed infinite dimensional subspace belongs to the same class, • inevitable, i.e. any Banach space contains an infinite dimensional subspace in one of those classes, • defined in terms of the richness of the family of bounded operators on/in the space. The program was inspired by the famous Gowers' dichotomy [15] exhibiting the first two classes: spaces with an unconditional basis and hereditary indecomposable spaces. Recall that a space is called hereditarily indecomposable (HI) if none of its closed infinite dimensional subspaces is a direct sum of its two closed infinite dimensional subspaces.The research now concentrates on identifying classes in terms of the family of isomorphisms defined in a space. The richness of this family can be stated in various "minimality" conditions, whereas the lack of certain type isomorphic embeddings of subspaces of a given space is described by different types of "tightness" of the considered space.Recall that a Banach space is minimal if it embeds isomorphically into any its closed infinite dimensional subspace. Relaxing this notion one obtains quasiminimality, which asserts that any two infinitely dimensional subspaces of a given space contain further two isomorphic infinitely dimensional subspaces. Relaxing the notions of minimality or adding additional requirements of choice of isomorphic subspaces in quasi-minimality case leads to different types of minimality of a space, contrasted in [16,10] with different types of tightness, categorized in [10].Recall that a subspace Y of a Banach space X with a basis (e n ) is tight in X if there is a sequence of successive subsets I 1 < I 2 < . . . of N such that the support of any isomorphic copy of Y in X intersects all but finitely many I n 's. X is called tight if any of its subspaces is tight in X. Adding requirements on the subsets (I n ) with 2000 Mathematics Subject Classification. 46B03.

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Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Types of tightness in spaces with unconditional basis

Manoussakis

Pelczar-Barwacz

2014

Studia Math.

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Types of tightness in spaces with unconditional basis

Manoussakis

Pelczar-Barwacz

2013

Preprint

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We present a reflexive Banach space with an unconditional basis which is quasi-minimal and tight by range, i.e. of type (4) in Ferenczi-Rosendal list within the framework of Gowers' classification program of Banach spaces, but contrary to the recently constructed space of type (4) also tight with constants, thus essentially extending the list of known examples in Gowers' program. The space is defined on the base on a boundedly modified mixed Tsirelson space with use of a special coding function.2000 Mathematics Subject Classification. 46B03.

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Quasi-minimality and tightness by range in spaces with unconditional basis

Cited by 2 publications

References 14 publications

Types of tightness in spaces with unconditional basis

Types of tightness in spaces with unconditional basis

Types of tightness in spaces with unconditional basis

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