Weak⁎-sequential properties of Johnson–Lindenstrauss spaces

Abstract

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.

On integration in banach spaces and total sets

“…Any Banach space with w * -angelic dual has property (D), see [22,Theorem 1]. The converse fails in general: this is witnessed by the Johnson-Lindenstrauss space JL 2 (F ) associated to any maximal almost disjoint family F of infinite subsets of N (see the introduction of [4] and the references therein). More generally, property (D ′ ) implies property (D), see [32,Proposition 12].…”

“…By the Banach-Dieudonné theorem, property (D ′ ) is formally weaker than property (E ′ ) of [27], which means that every w * -sequentially closed convex bounded subset of X * is w * -closed. The class of Banach spaces having property (E ′ ) includes those with w * -angelic dual as well as all spaces which are w * -sequentially dense in their bidual (see [4,Theorem 5.3]). On the other hand, we also study conditions ensuring the Γ-integrability of a Γ-scalarly integrable function.…”

𝐶*-algebras with weak* angelic dual balls

“…Indeed, it has been shown by Martínez-Cervantes [22] that the Johnson-Lindenstrauss space JL 2 has weak * sequential but not weak * angelic dual ball; that is, (b) ⇒ (a), which settled negatively a question of Plichko [25,Question 10]. See also [5] for further arguments on weak * sequential properties of JL 2 . As was also pointed out in [22], the space C([0, ω 1 ]) has weak * sequentially compact but not weak * sequential dual ball; and hence, (c) ⇒ (b).…”

On the range of a vector measure

“…The Banach space X is said to have Efremov's property (E) if the w * -closure of any convex set C ⊆ B X * consists of limits of w * -convergent sequences contained in C. Obviously, this property holds if (B X * , w * ) is Fréchet-Urysohn. Under the Continuum Hypothesis there exist Banach spaces separating both properties (see [2]), but it is unknown what happens in general. The relevance of property (E) to our discussion stems from the fact that if X has property (E), then (r(Z), w * ) has the Mazur property (see [9,Corollary 3.4]) and so does (Z, σ(Z, X)).…”