Abstract. A topological space is said to be sequential if every sequentially closed subspace is closed. We consider Banach spaces with weak*-sequential dual ball. In particular, we show that if X is a Banach space with weak*-sequentially compact dual ball and Y ⊂ X is a subspace such that Y and X/Y have weak*-sequential dual ball, then X has weak*-sequential dual ball. As an application we obtain that the Johnson-Lindenstrauss space JL 2 and C(K) for K scattered compact space of countable height are examples of Banach spaces with weak*-sequential dual ball, answering in this way a question of A. Plichko.

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We show that if a Banach lattice is projective, then every bounded sequence that can be mapped by a homomorphism onto the basis of c 0 must contain an ℓ 1 -subsequence. As a consequence, if Banach lattices ℓ p or F BL[E] are projective, then p = 1 or E has the Schur property, respectively. On the other hand, we show that C(K) is projective whenever K is an absolute neighbourhood retract, answering a question by de Pagter and Wickstead.2010 Mathematics Subject Classification. 46B43, 06BXX.

A Banach space X is said to have Efremov's property (E) if every element of the weak * -closure of a convex bounded set C ⊆ X * is the weak *limit of a sequence in C. By assuming the Continuum Hypothesis, we prove that there exist maximal almost disjoint families of infinite subsets of N for which the corresponding Johnson-Lindenstrauss spaces enjoy (resp. fail) property (E). This is related to a gap in [A. Plichko, Three sequential properties of dual Banach spaces in the weak * topology, Topology Appl. 190 (2015), [93][94][95][96][97][98] and allows to answer (consistently) questions of Plichko and Yost.

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.