Compactness in Banach space theory - selected problems

We list a number of problems in several topics related to compactness in nonseparable Banach spaces. Namely, about the Hilbertian ball in its weak topology, spaces of continuous functions on Eberlein compacta, WCG Banach spaces, Valdivia compacta and Radon-Nikod\'{y}m compacta

On Sikorski’s trace formula

The article contains no abstrac

Properties (u) and (V*) of Pelczynski in symmetric spaces of ? -measurable operators

It is shown that order continuity of the norm and weak sequential completeness in non-commutative strongly symmetric spaces of ? -measurable operators are respectively equivalent to properties (u) and (V?) of Pelczynski. In addition, it is shown that each strongly symmetric space with separable (Banach) bidual is necessarily reflexive. These results are non-commutative analogues of well-known characterisations in the setting of Banach lattices.Delft Institute of Applied MathematicsElectrical Engineering, Mathematics and Computer Scienc