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A Banach space dichotomy theorem for quotients of subspaces

Valentin Ferenczi (2007)

Studia Mathematica

A Banach space X with a Schauder basis is defined to have the restricted quotient hereditarily indecomposable property if X/Y is hereditarily indecomposable for any infinite-codimensional subspace Y with a successive finite-dimensional decomposition on the basis of X. The following dichotomy theorem is proved: any infinite-dimensional Banach space contains a quotient of a subspace which either has an unconditional basis, or has the restricted quotient hereditarily indecomposable property.

A Characterization of Weakly Lindelöf Determined Banach Spaces

Kalenda, Ondřej (2003)

Serdica Mathematical Journal

2000 Mathematics Subject Classification: 46B26, 46B03, 46B04.We prove that a Banach space X is weakly Lindelöf determined if (and only if) each non-separable Banach space isomorphic to a complemented subspace of X has a projectional resolution of the identity. This answers a question posed by S. Mercourakis and S. Negrepontis and yields a converse of Amir-Lindenstrauss’ theorem. We also prove that a Banach space of the form C(K) where K is a continuous image of a Valdivia compactum is weakly Lindelöf...

A C(K) Banach space which does not have the Schroeder-Bernstein property

Piotr Koszmider (2012)

Studia Mathematica

We construct a totally disconnected compact Hausdorff space K₊ which has clopen subsets K₊” ⊆ K₊’ ⊆ K₊ such that K₊” is homeomorphic to K₊ and hence C(K₊”) is isometric as a Banach space to C(K₊) but C(K₊’) is not isomorphic to C(K₊). This gives two nonisomorphic Banach spaces (necessarily nonseparable) of the form C(K) which are isomorphic to complemented subspaces of each other (even in the above strong isometric sense), providing a solution to the Schroeder-Bernstein problem for Banach spaces...

A Discretized Approach to W. T. Gowers' Game

V. Kanellopoulos, K. Tyros (2010)

Bulletin of the Polish Academy of Sciences. Mathematics

We give an alternative proof of W. T. Gowers' theorem on block bases by reducing it to a discrete analogue on specific countable nets. We also give a Ramsey type result on k-tuples of block sequences in a normed linear space with a Schauder basis.

A Gowers tree like space and the space of its bounded linear operators

Giorgos Petsoulas, Theocharis Raikoftsalis (2009)

Studia Mathematica

The famous Gowers tree space is the first example of a space not containing c₀, ℓ₁ or a reflexive subspace. We present a space with a similar construction and prove that it is hereditarily indecomposable (HI) and has ℓ₂ as a quotient space. Furthermore, we show that every bounded linear operator on it is of the form λI + W where W is a weakly compact (hence strictly singular) operator.

A New Hereditarily l^2 Banach Space

Petsoulas, Giorgos (2009)

Serdica Mathematical Journal

2000 Mathematics Subject Classification: 46B20, 46B26.We construct a non-reflexive, l^2 saturated Banach space such that every non-reflexive subspace has non-separable dual.

A new metric invariant for Banach spaces

F. Baudier, N. J. Kalton, G. Lancien (2010)

Studia Mathematica

We show that if the Szlenk index of a Banach space X is larger than the first infinite ordinal ω or if the Szlenk index of its dual is larger than ω, then the tree of all finite sequences of integers equipped with the hyperbolic distance metrically embeds into X. We show that the converse is true when X is assumed to be reflexive. As an application, we exhibit new classes of Banach spaces that are stable under coarse-Lipschitz embeddings and therefore under uniform homeomorphisms.

A note on extensions of Pełczyński's decomposition method in Banach spaces

Elói Medina Galego (2007)

Studia Mathematica

Let X,Y,A and B be Banach spaces such that X is isomorphic to Y ⊕ A and Y is isomorphic to X ⊕ B. In 1996, W. T. Gowers solved the Schroeder-Bernstein problem for Banach spaces by showing that X is not necessarily isomorphic to Y. In the present paper, we give a necessary and sufficient condition on sextuples (p,q,r,s,u,v) in ℕ with p + q ≥ 2, r + s ≥ 1 and u, v ∈ ℕ* for X to be isomorphic to Y whenever these spaces satisfy the following decomposition scheme: ⎧ X u X p Y q , ⎨ ⎩ Y v A r B s . Namely, Ω = (p-u)(s-r-v)...

A note on lattice renormings

Marián J. Fabián, Petr Hájek, Václav Zizler (1997)

Commentationes Mathematicae Universitatis Carolinae

It is shown that every strongly lattice norm on c 0 ( Γ ) can be approximated by C smooth norms. We also show that there is no lattice and Gâteaux differentiable norm on C 0 [ 0 , ω 1 ] .

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