Let be a simplicial function space on a metric compact space . Then the Choquet boundary of is an -set if and only if given any bounded Baire-one function on there is an -affine bounded Baire-one function on such that on . This theorem yields an answer to a problem of F. Jellett from [8] in the case of a metrizable set .
We introduce the notions of pointwise modulus of squareness and local modulus of squareness of a normed space X. This answers a question of C. Benítez, K. Przesławski and D. Yost about the definition of a sensible localization of the modulus of squareness. Geometrical properties of the norm of X (Fréchet smoothness, Gâteaux smoothness, local uniform convexity or strict convexity) are characterized in terms of the behaviour of these moduli.
A new drop property, the quasi-weak drop property, is introduced. Using streaming sequences introduced by Rolewicz, a characterisation of the quasi-weak drop property is given for closed bounded convex sets in a Fréchet space. From this, it is shown that the quasi-weak drop property is equivalent to weak compactness. Thus a Fréchet space is reflexive if and only if every closed bounded convex set in the space has the quasi-weak drop property.
Every weakly sequentially compact convex set in a locally convex space has the weak drop property and every weakly compact convex set has the quasi-weak drop property. An example shows that the quasi-weak drop property is strictly weaker than the weak drop property for closed bounded convex sets in locally convex spaces (even when the spaces are quasi-complete). For closed bounded convex subsets of quasi-complete locally convex spaces, the quasi-weak drop property is equivalent to weak compactness....
Let and be algebras of subsets of a set with , and denote by the set of all quasi-measure extensions of a given quasi-measure on to . We give some criteria for order boundedness of in , in the general case as well as for atomic . Order boundedness implies weak compactness of . We show that the converse implication holds under some assumptions on , and or alone, but not in general.
For a continuous map T of a compact metrizable space X with finite topological entropy, the order of accumulation of entropy of T is a countable ordinal that arises in the context of entropy structures and symbolic extensions. We show that every countable ordinal is realized as the order of accumulation of some dynamical system. Our proof relies on functional analysis of metrizable Choquet simplices and a realization theorem of Downarowicz and Serafin. Further, if M is a metrizable Choquet simplex,...
Let and be algebras of subsets of a set with , and denote by the set of all quasi-measure extensions of a given quasi-measure on to . We show that is order bounded if and only if it is contained in a principal ideal in if and only if it is weakly compact and is contained in a principal ideal in . We also establish some criteria for the coincidence of the ideals, in , generated by and .