Weak-star point of continuity property and Schauder bases (Q2866768)

\(\mathbb N^{<\omega}\) denotes the set of all finite sequences in \(\mathbb N\) ordered by extension. Let \(X\) be a Banach space. A family \(\{x_A\}\), \(A\in \mathbb N^{<\omega}\), of elements of \(X^*\) is called a topologically weakly* null tree if 0 belongs to the weak* closure of the set \(\{x_{(A,n)}\mid n=1,2,\dots\}\) for every \(A\in \mathbb N^{<\omega}\). The tree \(\{x_A\}\) is called w*-null if the sequence \(\{x_{(A,n)}\mid n=1,2,\dots\}\) is w*-null for every \(A\in \mathbb N^{<\omega}\). A branch of a tree \(\{x_A\}\) is a sequence \(\{x_{\emptyset}, x_{(n_1)},x_{(n_1,n_2)},\dots\}\) for some sequence \(n_1,n_2,n_3,\ldots \in\mathbb N\). A bounded subset \(K\) of the Banach space \(X^*\) has the weak-star point of continuity property (w*PCP) if every non-empty subset of \(K\) has relatively w*-open subsets of arbitrarily small diameter. A sequence \(\{e_n\}\) in a Banach space is said to be of type \(P\) if the set \(\{\sum_{k=1}^n e_k\mid n=1,2,\dots\}\) is bounded. In this paper, the authors show the following results:NEWLINENEWLINEProposition 2.1. Let \(K\) be a w*-compact and convex subset of a Banach space \(X^*\). The following statements are equivalent: {\parindent=6mm \begin{itemize}\item[(1)] \(K\) fails the Radon-Nikodým property (RNP). \item[(2)] There exists a seminormalized topologically weak* null tree \(\{x_A\}\), \(A\in \mathbb N^{<\omega}\), of elements of \(X^*\) such that the set \(\{\sum_{B<A} x_B \mid A\in \mathbb N^{<\omega}\}\) is contained in \(K\). NEWLINENEWLINE\end{itemize}} Proposition 2.2. Let \(X\) be a separable Banach space and let \(K\) be a closed and bounded subset of \(X^*\). The following are equivalent: {\parindent=6mm \begin{itemize}\item[(1)] \(K\) fails the w*PCP. \item[(2)] There exists a seminormalized weak* null tree \(\{x_A\}\), \(A\in \mathbb N^{<\omega}\), of elements of \(X^*\) such that the set \(\{\sum_{B<A} x_B\mid A\in \mathbb N^{<\omega}\}\) is contained in \(K\). NEWLINENEWLINE\end{itemize}} The authors prove the following characterization of the w*PCP:NEWLINENEWLINETheorem 2.4. Let \(Y\) be a separable Banach space and \(X\) a subspace of \(Y^*\). The following statements are equivalent: {\parindent=8mm \begin{itemize}\item[(i)] The unit ball of \(X\) has the w*PCP. \item[(ii)] No w*-null tree in the unit sphere of \(X\) is uniformly type \(P\). \item[(iii)] No w*-null tree in the unit sphere of \(X\) has a type \(P\) branch. \item[(iv)] Every w*-null tree in the unit sphere of \(X\) has a boundedly complete branch. NEWLINENEWLINE\end{itemize}} The authors also prove the following theorem.NEWLINENEWLINETheorem 2.6. Let \(X\) be a Banach space. TFAE: {\parindent=8mm \begin{itemize}\item[(i)] \(X^*\) has the RNP. \item[(ii)] No topologically w*-null tree in the unit sphere of \(X\) is uniformly type \(P\). \item[(iii)] No topologically w*-null tree in the unit sphere of \(X\) has a boundedly complete branch. NEWLINENEWLINE\end{itemize}} As a corollary, the following result of H. P. Rosenthal is obtained: If the Banach space \(X\) has the point of continuity property (PCP), then every semi-normalized basic sequence in \(X\) has a boundedly complete subsequence.NEWLINENEWLINEThe following is an important open problem: If every subspace of \(X\) with a basis has the RNP, does this imply that \(X\) has the RNP? The authors show that the dual of a separable space has the RNP iff every subspace with a basis has the w*PCP.