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An Example Concerning Valdivia Compact Spaces

Kalenda, Ondrej — 1999

Serdica Mathematical Journal

∗ Supported by Research grants GAUK 190/96 and GAUK 1/1998 We prove that the dual unit ball of the space C0 [0, ω1 ) endowed with the weak* topology is not a Valdivia compact. This answers a question posed to the author by V. Zizler and has several consequences. Namely, it yields an example of an affine continuous image of a convex Valdivia compact (in the weak* topology of a dual Banach space) which is not Valdivia, and shows that the property of the dual unit ball being Valdivia is...

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...

Continuous images and other topological properties of Valdivia compacta

Ondřej Kalenda — 1999

Fundamenta Mathematicae

We study topological properties of Valdivia compact spaces. We prove in particular that a compact Hausdorff space K is Corson provided each continuous image of K is a Valdivia compactum. This answers a question of M. Valdivia (1997). We also prove that the class of Valdivia compacta is stable with respect to arbitrary products and we give a generalization of the fact that Corson compacta are angelic.

Valdivia compacta and equivalent norms

Ondřej Kalenda — 2000

Studia Mathematica

We prove that the dual unit ball of a Banach space X is a Corson compactum provided that the dual unit ball with respect to every equivalent norm on X is a Valdivia compactum. As a corollary we show that the dual unit ball of a Banach space X of density 1 is Corson if (and only if) X has a projectional resolution of the identity with respect to every equivalent norm. These results answer questions asked by M. Fabian, G. Godefroy and V. Zizler and yield a converse to Amir-Lindenstrauss’ theorem.

Rich families and elementary submodels

Marek CúthOndřej 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...

(I)-envelopes of unit balls and James' characterization of reflexivity

Ondřej F. K. Kalenda — 2007

Studia Mathematica

We study the (I)-envelopes of the unit balls of Banach spaces. We show, in particular, that any nonreflexive space can be renormed in such a way that the (I)-envelope of the unit ball is not the whole bidual unit ball. Further, we give a simpler proof of James' characterization of reflexivity in the nonseparable case. We also study the spaces in which the (I)-envelope of the unit ball adds nothing.

Complex Banach spaces with Valdivia dual unit ball.

Ondrej F. K. Kalenda — 2005

Extracta Mathematicae

We study the classes of complex Banach spaces with Valdivia dual unit ball. We give complex analogues of several theorems on real spaces. Further we study relationship of these complex Banach spaces with their real versions and that of real Banach spaces and their complexification. We also formulate several open problems.

A converse to Amir-Lindenstrauss theorem in complex Banach spaces.

Ondrej F. K. Kalenda — 2006

RACSAM

We show that a complex Banach space is weakly Lindelöf determined if and only if the dual unit ball of any equivalent norm is weak* Valdivia compactum. We deduce that a complex Banach space X is weakly Lindelöf determined if and only if any nonseparable Banach space isomorphic to a complemented subspace of X admits a projectional resolution of the identity. These results complete the previous ones on real spaces.

Remark on the Abstract Dirichlet Problem for Baire-One Functions

Ondřej F. K. Kalenda — 2005

Bulletin of the Polish Academy of Sciences. Mathematics

We study the possibility of extending any bounded Baire-one function on the set of extreme points of a compact convex set to an affine Baire-one function and related questions. We give complete solutions to these questions within a class of Choquet simplices introduced by P. J. Stacey (1979). In particular we get an example of a Choquet simplex such that its set of extreme points is not Borel but any bounded Baire-one function on the set of extreme points can be extended to an affine Baire-one function....

Baire-one mappings contained in a usco map

Ondřej F. K. Kalenda — 2007

Commentationes Mathematicae Universitatis Carolinae

We investigate Baire-one functions whose graph is contained in the graph of a usco mapping. We prove in particular that such a function defined on a metric space with values in d is the pointwise limit of a sequence of continuous functions with graphs contained in the graph of a common usco map.

On projectional skeletons in Vašák spaces

Ondřej F. K. Kalenda — 2017

Commentationes Mathematicae Universitatis Carolinae

We provide an alternative proof of the theorem saying that any Vašák (or, weakly countably determined) Banach space admits a full 1 -projectional skeleton. The proof is done with the use of the method of elementary submodels and is comparably simple as the proof given by W. Kubiś (2009) in case of weakly compactly generated spaces.

Embedding of the ordinal segment [ 0 , ω 1 ] into continuous images of Valdivia compacta

Ondřej F. K. Kalenda — 1999

Commentationes Mathematicae Universitatis Carolinae

We prove in particular that a continuous image of a Valdivia compact space is Corson provided it contains no homeomorphic copy of the ordinal segment [ 0 , ω 1 ] . This generalizes a result of R. Deville and G. Godefroy who proved it for Valdivia compact spaces. We give also a refinement of their result which yields a pointwise version of retractions on a Valdivia compact space.

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