Banach spaces without minimal subspaces

Ferenczi, Valentin; Rosendal, Christian

doi:10.1016/j.jfa.2009.01.028

Cited by 38 publications

(160 citation statements)

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“…This was further refined in [3] and, in [14], was formulated as the determinacy of two related adversarial Gowers games, A X and B X .…”

Section: Introductionmentioning

confidence: 99%

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Determinacy of adversarial Gowers games

Rosendal

2014

Fund. Math.

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Abstract. We prove a game theoretic dichotomy for G δσ sets of block sequences in vector spaces that extends, on the one hand, the block Ramsey theorem of W. T. Gowers proved for analytic sets of block sequences and, on the other hand, M. Davis' proof of Σ 0 3 determinacy. IntroductionIn the present paper, we prove an extension of W. T. Gowers' Ramsey theorem for block sequences in normed vector spaces [6]. This was instrumental in the proof of his dichotomy for Banach spaces between containing an unconditional basic sequence or a hereditarily indecomposable subspace that ultimately led to a solution of the homogeneous space problem for Banach spaces.The statement of Gowers' theorem is as follows. Assume that A is an analytic set of sequences (y n ) of normalised vectors in a separable Banach space E and, moreover, any infinite-dimensional subspace X ⊆ E contains a sequence from A. Then there is an infinite-dimensional subspace X ⊆ E for which one can sequentially choose the terms of some (y n ) close to A such that the y n belong to any given infinite-dimensional subspaces Y n ⊆ X.The precise statement is formulated in terms of a game in which player I plays the subspaces Y n ⊆ X, while player II choses the vectors y n ∈ Y n . While we shall not follow Gowers' lead in dealing with normed vector spaces, but instead use the set-up of [14] and thus consider only vector spaces over countable fields, the exact results proved here easily imply slightly stronger, but approximate, statements for normed vector spaces as is shown in [14].So suppose that E is a countable-dimensional vector space over a countable field F. We define the Gowers game G X played below an infinite-dimensional subspace X ⊆ E as follows. Players I and II alternate in playing respectively infinite-dimensional subspaces Y n ⊆ X and non-zero vectors y n ∈ Y n ,Similarly, the infinite asymptotic game F X is defined as the Gowers game except that I is now required to play subspaces Y n of finite codimension in X (in fact, even so-called tail subspaces). Thus, from the viewpoint of II, the game has not changed, but, in F X , player I will have significantly less control over where player II chooses 2000 Mathematics Subject Classification. Primary: 46B03, Secondary 03E15.

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“…This was further refined in [3] and, in [14], was formulated as the determinacy of two related adversarial Gowers games, A X and B X .…”

Section: Introductionmentioning

confidence: 99%

“…For applications of the above dichotomies to the geometry of Banach spaces, we refer the reader to [6], [3] and [13]. Acknowledgement: The author is grateful for a number of insightful comments and useful discussions with A. Montalban, J. Moore and P. Welch on the topic of this paper.…”

Section: Introductionmentioning

confidence: 99%

Determinacy of adversarial Gowers games

Rosendal

2014

Fund. Math.

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“…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].…”

Section: Introductionmentioning

confidence: 99%

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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“…(6) No disjointly supported subspaces of X wh are isomorphic. Using the terminology from [14] the space X wh is tight by support. In the above and herein, we use L(Y ) to denote the Banach space of bounded linear operators on Y .…”

Section: Introductionmentioning

confidence: 99%

“…In the next section we give a description of the norm of X wh and further discuss some of its critical properties. We note that in [14] they construct a space that is strongly asymptotic 1 and has a property they call tight by support. The space X wh is the first example of a strongly asymptotic 2 space that is tight by support.…”

Section: Introductionmentioning

confidence: 99%