Ondřej F. K. Kalenda

Stegall compact spaces which are not fragmentable

Abstract Using modifications of the well-known construction of “double-arrow” space we give consistent examples of nonfragmentable compact Hausdorff spaces which belong to Stegalls class S . Namely the following is proved. (1) If ℵ 1 is less than the least inaccessible cardinal in L and MA & ¬ CH hold then there is a nonfragmentable compact Hausdorff space K such that every minimal usco mapping of a Baire space into K is singlevalued at points of a residual set. (2) If V=L then there is a nonfragmentable compact Hausdorff space K such that every minimal usco mapping of a completely regular Baire space into K is singlevalued at points of a residual set.

Quantitative Dunford–Pettis property

Abstract We investigate possible quantifications of the Dunford–Pettis property. We show, in particular, that the Dunford–Pettis property is automatically quantitative in a sense. Further, there are two incomparable mutually dual stronger versions of a quantitative Dunford–Pettis property. We prove that L 1 spaces and C ( K ) spaces possess both of them. We also show that several natural measures of weak non-compactness are equal in L 1 spaces.

ON QUANTIFICATION OF WEAK SEQUENTIAL COMPLETENESS

We consider several quantities related to weak sequential completeness of a Banach space and prove some of their properties in general and in L-embedded Banach spaces, improving in particular an inequality of G. Godefroy, N. Kalton and D. Li. We show some examples witnessing natural limits of our positive results, in particular, we construct a separable Banach space X with the Schur property that cannot be renormed to have a certain quantitative form of weak sequential completeness, thus providing a partial answer to a question of G. Godefroy.

Complementation in spaces of continuous functions on compact lines

Abstract We characterize order preserving continuous surjections between compact linearly ordered spaces which admit an averaging operator, together with estimates of the norm of such an operator. This result is used to the study of strengthenings of the separable complementation property in spaces of continuous functions on compact lines. These properties include in particular continuous separable complementation property and existence of a projectional skeleton.

Monotone retractability and retractional skeletons

Abstract We prove that a countably compact space is monotonically retractable if and only if it has a full retractional skeleton. In particular, a compact space is monotonically retractable if and only if it is Corson. This gives an answer to a question of R. Rojas-Hernandez and V.V. Tkachuk. Further, we apply this result to characterize retractional skeleton using a topology on the space of continuous functions, answering thus a question of the first author and a related question of W. Kubiś.

On a difference between quantitative weak sequential completeness and the quantitative Schur property

We study quantitative versions of the Schur property and weak sequential completeness, proceeding thus with investigations started by G. Godefroy, N. Kalton and D. Li and continued by H. Pfitzner and the authors. We show that the Schur property of

Quantification of the reciprocal Dunford-Pettis property

\ell_1

The structure of Valdivia compact lines

holds quantitatively in the strongest possible way and construct an example of a Banach space which is quantitatively weakly sequentially complete, has the Schur property but fails the quantitative form of the Schur property.

Decompositions of preduals of JBW and JBW⁎ algebras

We prove in particular that Banach spaces of the form

On Markushevich bases in preduals of von Neumann algebras

C_0(\Omega)