On weakly Radon–Nikodým compact spaces

Abstract

Abstract. A compact space is said to be weakly Radon-Nikodým if it is homeomorphic to a weak * -compact subset of the dual of a Banach space not containing an isomorphic copy of ℓ 1 . In this paper we provide an example of a continuous image of a Radon-Nikodým compact space which is not weakly Radon-Nikodým. Moreover, we define a superclass of the continuous images of weakly Radon-Nikodým compact spaces and study its relation with Corson compacta and weakly Radon-Nikodým compacta.In [9], I. Namioka defined a compact space K to be Radon-Nikodým (RN for short) if and only if it is homeomorphic to a weak * -compact subset of a dual Banach space with the Radon-Nikodým property. One of the most important questions posed by Namioka in this paper was whether the class of RN compact spaces is closed under continuous images. This question was solved negatively by A. Avilés and P. Koszmider in [3].The class of weakly Radon-Nikodým compact spaces generalizes the class of RN compact spaces. In [7], E. Glasner and M. Megrelishvili define a compact space to be weakly Radon-Nikodým (WRN for short) if and only if it is homeomorphic to a weak * -compact subset of the dual of a Banach space not containing an isomorphic copy of ℓ 1 . In [8], they ask whether the class of WRN compact spaces is stable under continuous images. In these papers, a picture of this class of compact spaces is given and it is characterized in terms of Banach spaces of continuous functions. They prove that linearly ordered compact spaces are WRN. Moreover, [8] contains a proof by S. Todorcevic that βN is not WRN.In this work we answer negatively the question of E. Glasner and M. Megrelishvili by proving that a modification of the construction given in [3] provides an example of a continuous image of a RN compact space which is not WRN. We construct a RN compact space L 0 , a non-WRN compact space L 1 and a surjective continuous function π : L 0 −→ L 1 in the same way as in [3].In the second section we define quasi WRN compact spaces, a superclass of the continuous images of WRN compact spaces. We prove that zero-dimensional quasi WRN compact spaces are WRN. We also show other relations of quasi WRN compacta with Corson and Eberlein compacta, including an example of a Corson compact space which is not quasi WRN.

Open problems in Banach spaces and measure theory

“…Some results on these spaces can also be found in [114,Section 2.3]. While preparing his PhD Thesis, G. Martínez-Cervantes [89] is studying WPG spaces and the class of weak Radon-Nikodým (WRN) compacta introduced in [64]. A compact K is said to be WRN if it is homeomorphic to a w * -compact subset of the dual of a Banach space not containing ℓ 1 ; this condition is equivalent to C(K) being WPG.…”

Circularly ordered dynamical systems

“…Note that βN is not WRN, a result of Todorcević (see [21]). Another result from [33], shows that the Talagrand's compact is also not WRN. Although G is representable on a (separable) Rosenthal Banach space, we have Asp(G) = {constants} and therefore any Asplund representation of this group is trivial (this situation is similar to the case of the group H + [0, 1], [17]).…”

“…In a recent paper [33] Martinez-Cervantes shows that a continuous image of a WRN compact space need not be WRN. This answers a question from [21].…”

On Fragmentable Compact Lines

“…A natural class which generalizes both the class of linearly ordered compact spaces and the class of RN compact spaces is the class of weakly Radon-Nikodým (WRN) compacta (see [8] for the definition). Nevertheless, S. Argyros proved the existence of Gul'ko WRN compact spaces which are not Eberlein (see [1] and [7, Section 2.4]).…”