Summing Boolean algebras
Let \(X\) be a Banach space. The classical Orlicz-Pettis theorem says that weak subseries convergence already implies subseries convergence. The authors use the term \(P(\mathbb N)\) is \(X\)-weakly summing to express that the conclusion of the Orlicz-Pettis theorem holds in \(X\) when subseries corresponding to all subsets of \(\mathbb N\) are considered and \(P(\mathbb N)\) is weakly summing to say that this is true for every \(X\). An immediate question is to ask how many subseries must be looked at in order to obtain the conclusion of the Orlicz-Pettis theorem. In other words: Which subfamilies of \(P(\mathbb N)\) are weakly \(X\)-summing? Posing the question this way opens the possibility of having different subfamilies for different \(X\). The question has been considered by many authors and it is a well-known result that property \(FQ\) of a natural family (a subfamily of \(P(\mathbb N)\) containing all finite subsets of \(\mathbb N\)) or subsequential completeness of a Boolean algebra ensures it is weakly summing. Very interesting is the result (Theorem 3.19) that a natural Boolean algebra is weakly summing if and only if it is \(c_0\)-weakly summing. Theorem 3.14, which perhaps is the main result, states that every natural family with property \(S_1\) is weakly summing. The family \(\mathcal F\subset P(\mathbb N)\) has property \(S_1\) if for every pair \([(A_i)_i,(B_i)_i]\) of disjoint sequences of mutually disjoint finite sets, there are an infinite \(M\subset\mathbb N\) and a \(B\in \mathcal F\) such that \(A_i\subset B\) and \(B_i\subset B^C\) for all \(i\in M\). The authors give examples to show that both \(FQ\) and subsequential completeness are stronger than \(S_1\). A key tool in the paper is the relationship between algebraic properties of Boolean algebras and topological properties of the corresponding Stone spaces. New in the paper is the concept of a successional subalgebra, which enables one to associate any sequence of mutually disjoint sets with a sequence of atoms in its Stone representation.
scientific article; zbMATH DE number 2174208
scientific article