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Perturbations of isometries between C(K)-spaces

Yves Dutrieux, Nigel J. Kalton (2005)

Studia Mathematica

We study the Gromov-Hausdorff and Kadets distances between C(K)-spaces and their quotients. We prove that if the Gromov-Hausdorff distance between C(K) and C(L) is less than 1/16 then K and L are homeomorphic. If the Kadets distance is less than one, and K and L are metrizable, then C(K) and C(L) are linearly isomorphic. For K and L countable, if C(L) has a subquotient which is close enough to C(K) in the Gromov-Hausdorff sense then K is homeomorphic to a clopen subset of L.

Pointwise compactness and continuity of the integral.

G. Vera (1996)

Revista Matemática de la Universidad Complutense de Madrid

In this paper we bring together the different known ways of establishing the continuity of the integral over a uniformly integrable set of functions endowed with the topology of pointwise convergence. We use these techniques to study Pettis integrability, as well as compactness in C(K) spaces endowed with the topology of pointwise convergence on a dense subset D in K.

Quotients of indecomposable Banach spaces of continuous functions

Rogério Augusto dos Santos Fajardo (2012)

Studia Mathematica

Assuming ⋄, we construct a connected compact topological space K such that for every closed L ⊂ K the Banach space C(L) has few operators, in the sense that every operator on C(L) is multiplication by a continuous function plus a weakly compact operator. In particular, C(K) is indecomposable and has continuum many non-isomorphic indecomposable quotients, and K does not contain a homeomorphic copy of βℕ. Moreover, assuming CH we construct a connected compact K where C(K) has few...

Radon-Nikodým compact spaces of low weight and Banach spaces

Antonio Avilés (2005)

Studia Mathematica

We prove that a continuous image of a Radon-Nikodým compact of weight less than b is Radon-Nikodým compact. As a Banach space counterpart, subspaces of Asplund generated Banach spaces of density character less than b are Asplund generated. In this case, in addition, there exists a subspace of an Asplund generated space which is not Asplund generated and which has density character exactly b.

Regular Averaging and Regular Extension Operators in Weakly Compact Subsets of Hilbert Spaces

Argyros, Spiros, Arvanitakis, Alexander (2004)

Serdica Mathematical Journal

2000 Mathematics Subject Classification: Primary 46E15, 54C55; Secondary 28B20.For weakly compact subsets of Hilbert spaces K, we study the existence of totally disconnected spaces L, such that C(K) is isomorphic to C(L). We prove that the space C(BH ) admits a Pełczyński decomposition and we provide a starshaped weakly compact K, subset of BH with non-empty interior in the norm topology, and such that C(K) ~= C(L) with L totally disconnected.Research partially supported by EPEAEK program “Pythagoras”....

Rich families and elementary submodels

Marek Cúth, Ondřej Kalenda (2014)

Open Mathematics

We compare two methods of proving separable reduction theorems in functional analysis - the method of rich families and the method of elementary submodels. We show that any result proved using rich families holds also when formulated with elementary submodels and the converse is true in spaces with fundamental minimal system and in spaces of density ℵ1. We do not know whether the converse is true in general. We apply our results to show that a projectional skeleton may be without loss of generality...

Separable reduction theorems by the method of elementary submodels

Marek Cúth (2012)

Fundamenta Mathematicae

We simplify the presentation of the method of elementary submodels and we show that it can be used to simplify proofs of existing separable reduction theorems and to obtain new ones. Given a nonseparable Banach space X and either a subset A ⊂ X or a function f defined on X, we are able for certain properties to produce a separable subspace of X which determines whether A or f has the property in question. Such results are proved for properties of sets: of being dense, nowhere dense, meager, residual...

Sequential closedness of Boolean algebras of projections in Banach spaces

D. H. Fremlin, B. de Pagter, W. J. Ricker (2005)

Studia Mathematica

Complete and σ-complete Boolean algebras of projections acting in a Banach space were introduced by W. Bade in the 1950's. A basic fact is that every complete Boolean algebra of projections is necessarily a closed set for the strong operator topology. Here we address the analogous question for σ-complete Boolean algebras: are they always a sequentially closed set for the strong operator topology? For the atomic case the answer is shown to be affirmative. For the general case, we develop criteria...

Smoothness in Banach spaces. Selected problems.

Marian Fabian, Vicente Montesinos, Václav Zizler (2006)

RACSAM

This is a short survey on some recent as well as classical results and open problems in smoothness and renormings of Banach spaces. Applications in general topology and nonlinear analysis are considered. A few new results and new proofs are included. An effort has been made that a young researcher may enjoy going through it without any special pre-requisites and get a feeling about this area of Banach space theory. Many open problems of different level of difficulty are discussed. For the reader...

Some examples of continuous images of Radon-Nikodým compact spaces

Alexander D. Arvanitakis, Antonio Avilés (2009)

Czechoslovak Mathematical Journal

We provide a characterization of continuous images of Radon-Nikodým compacta lying in a product of real lines and model on it a method for constructing natural examples of such continuous images.

Some remarks on Radon-Nikodym compact spaces

Alexander D. Arvanitakis (2002)

Fundamenta Mathematicae

The class of quasi Radon-Nikodým compact spaces is introduced. We prove that this class is closed under countable products and continuous images. It includes the Radon-Nikodým compact spaces. Adapting Alster's proof we show that every quasi Radon-Nikodým and Corson compact space is Eberlein. This generalizes earlier results by J. Orihuela, W. Schachermayer, M. Valdivia and C. Stegall. Further the class of almost totally disconnected spaces is defined and it is shown that every quasi Radon-Nikodým...

Some remarks on universality properties of ℓ ∞ / c ₀

Mikołaj Krupski, Witold Marciszewski (2012)

Colloquium Mathematicae

We prove that if is not a Kunen cardinal, then there is a uniform Eberlein compact space K such that the Banach space C(K) does not embed isometrically into ℓ ∞ / c ₀ . We prove a similar result for isomorphic embeddings. Our arguments are minor modifications of the proofs of analogous results for Corson compacta obtained by S. Todorčević. We also construct a consistent example of a uniform Eberlein compactum whose space of continuous functions embeds isomorphically into ℓ ∞ / c ₀ , but fails to embed isometrically....

Strong subdifferentiability of norms and geometry of Banach spaces

Gilles Godefroy, Vicente Montesinos, Václav Zizler (1995)

Commentationes Mathematicae Universitatis Carolinae

The strong subdifferentiability of norms (i.eȯne-sided differentiability uniform in directions) is studied in connection with some structural properties of Banach spaces. It is shown that every separable Banach space with nonseparable dual admits a norm that is nowhere strongly subdifferentiable except at the origin. On the other hand, every Banach space with a strongly subdifferentiable norm is Asplund.

The controlled separable projection property for Banach spaces

Jesús Ferrer, Marek Wójtowicz (2011)

Open Mathematics

Let X, Y be two Banach spaces. We say that Y is a quasi-quotient of X if there is a continuous operator R: X → Y such that its range, R(X), is dense in Y. Let X be a nonseparable Banach space, and let U, W be closed subspaces of X and Y, respectively. We prove that if X has the Controlled Separable Projection Property (CSPP) (this is a weaker notion than the WCG property) and Y is a quasi-quotient of X, then the structure of Y resembles the structure of a separable Banach space: (a) Y/W is norm-separable...

The Dual of a Non-reflexive L-embedded Banach Space Contains l ∞ Isometrically

Hermann Pfitzner (2010)

Bulletin of the Polish Academy of Sciences. Mathematics

A Banach space is said to be L-embedded if it is complemented in its bidual in such a way that the norm between the two complementary subspaces is additive. We prove that the dual of a non-reflexive L-embedded Banach space contains l ∞ isometrically.

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