Compactness in Banach space theory —selected problems

Abstract

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-Nikodým compacta. Compacidad en espacios de Banach -problemas escogidosResumen. Enumeramos una serie de problemas en diferentes temas relacionados con compacidad en espacios de Banach no separables. Concretamente, sobra la bola euclídea en su topología débil, espacios de funciones continuas en compactos de Eberlein, espacios de Banach débilmente compactamente generados, compactos de Valdivia y compactos de Radon-Nikodým.Problem 24 Let K be an RN compact space. Is K homeomorphic to a subspace of P (S) for S scattered compact? Is it a subspace of M 1 (S)? Is it a continuous image of a subspace of M 1 (S)?Problem 25 Let K be a RN compact (or quasi RN compact) space. Is K the continuous image of a zero-dimensional RN compact space L? Can a continuous surjection f : L −→ K be obtained with an averaging operator?

Simultaneous projectional skeletons

“…This is a contradiction with the choice of M. The following question has already been articulated in [10] and [1]. Question 5.6.…”

“…We put Σ(Γ) = {x ∈ R Γ : |{γ ∈ Γ : x(γ) = 0}| ≤ ω}. Given a compact K, A ⊂ K is called a Σ-subset of K if there is a homeomorphic embedding h : K → [0, 1] κ such that A = h −1 [Σ(κ)]. A compact space K is said to be Valdivia compact if there exists a dense Σ-subset of K.…”

A continuous image of a Radon–Nikodým compact space which is not Radon–Nikodým

“…Other articles devoted to the problem of the continuous image include [4,14,19,25]. More information can be found in [7,12,13,26], which are dedicated to the topic, or contain sections dedicated to it. The purpose of this article is to provide a negative solution to the general problem: Theorem 1.1.…”

“…The compact spaces that we construct have weight c but we do not know if perhaps b is the optimal weight of a counterexample to the problem. The reader can find in [7] and [13] a number of interesting problems on RN compacta that still remain open. For example, we may mention that it is unknown if every RN compact space is the continuous image of a zero-dimensional RN compact space, or if it is always homeomorphic to a subspace of the space of probability measures on a scattered space.…”

Universal objects and associations between classes of Banach spaces and classes of compact spaces

“…This lack of absoluteness is known to be unavoidable as it is shown in most cases that some other axioms or forcing arguments imply the nonexistence of the constructions. This is related to Problem 24 of [7] where among others it is asked if every Radon-Nikodým compact space appears as a subspace of the dual ball B C(K) * where K is a scattered compact. Our questions seems to be a much weaker version of this question.…”