A chain condition for operators from \(C(K)\)-spaces (Q2922445)

If \(K\) is a compact Hausdorff space and \(X\) is an arbitrary Banach space, a well-known theorem of Pełczyński says that an operator \(T : C (K) \to X\) is weakly compact if and only if it does not fix a copy of \(c_0\) (meaning that there is no subspace \(Y\) of \(C (K)\) isomorphic to \(c_0\) on which \(T\) is an isomorphism from \(Y\) onto \(T (Y)\)).NEWLINENEWLINEIn the paper under review, the authors consider the following ordering on \(C (K)\): they say that \(f \prec g\) if \(f (x) = g (x)\) for every \(x \in \text{supp}\, (f)\).NEWLINENEWLINENoting that, if \((f_n)_{n = 1}^\infty\) is a bounded disjointly supported sequence of functions, then \(F = \{ \sum_{k = 1}^n f_k \, ; \;n \geq 1 \}\) is a bounded infinite \(\prec\)-chain, they prove that the map \(T : C (K) \to X\) is weakly compact if and only if, for each bounded infinite \(\prec\)-chain \(F \subseteq C (K)\), we have \(\inf \{ \| T f - T g\| \, ; \;f, g \in F, f \neq g\} = 0\).NEWLINENEWLINEThat leads them to consider the following property \((\star)\) for \(T : C (K) \to X\):NEWLINENEWLINE For each bounded uncountable \(\prec\)-chain \(F\) in \(C (K)\): \(\inf \{\| Tf - Tg\| : f, g \in F, \;f \neq g\} = 0\).NEWLINENEWLINE It follows that every weakly compact operator \(T\) satisfies \((\star)\).NEWLINENEWLINEThey prove that the converse holds when \(K\) is extremely disconnected.NEWLINENEWLINEThen, the authors investigate when the identity map on \(C (K)\) has property \((\star)\). They prove that this is the case when \(K\) is countable, when \(K\) is the one-point compactification of a discrete space, and when \(K\) is locally connected and satisfies the condition \(c.c.c.\) (meaning that the cardinality of families of pairwise disjoint non-void open sets is countable).NEWLINENEWLINEThey also prove that the set of operators \(T : C (K) \to C (K)\) satisfying \((\star)\) is a closed left ideal in the space of all operators on \(C (K)\).