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

We introduce a chain condition (bishop), defined for operators acting on C(K)-spaces, which is intermediate between weak compactness and having weakly compactly generated range. It is motivated by Pe{\l}czy'nski’s characterisation of weakly compact operators on C(K)-spaces. We prove that if K is extremally disconnected and X is a Banach space then an operator T : C(K) -> X is weakly compact if and only if it satisfies (bishop) if and only if the representing vector measure of T satisfies an analogous chain condition. As a tool for proving the above-mentioned result, we derive a topological counterpart of Rosenthal’s lemma. We exhibit several compact Hausdorff spaces K for which the identity operator on C(K) satisfies (bishop), for example both locally connected compact spaces having countable cellularity and ladder system spaces have this property. Using a Ramsey-type theorem, due to Dushnik and Miller, we prove that the collection of operators on a C(K)-space satisfying (bishop) forms a closed left ideal of B(C(K)).

💡 Research Summary

The paper introduces a new chain condition, denoted (bishop), for linear operators acting on spaces of continuous functions C(K). The condition sits strictly between weak compactness and the property that the operator’s range is weakly compactly generated (WCG). Formally, for a bounded linear operator T : C(K) → X, (bishop) requires that for every increasing net (or chain) {fα}α∈A of functions in C(K) which has a supremum, the images {T fα} form a Cauchy net in X; equivalently, given any ε > 0 there exist indices α ≤ β such that ‖T fα – T fβ‖ < ε. This is weaker than the classical definition of weak compactness (which demands that every bounded net has a weakly convergent subnet) but stronger than merely requiring that the range be contained in a WCG subspace.

The main results are proved under the hypothesis that the compact Hausdorff space K is extremally disconnected (the closure of every open set is open). In this setting the authors establish three equivalent statements:

1. T is weakly compact.
2. T satisfies the (bishop) chain condition.
3. The representing vector measure μT associated with T satisfies an analogous chain condition on the σ‑algebra of Borel subsets of K.

The equivalence (1) ⇔ (2) is obtained by adapting Pełczyński’s classical characterisation of weakly compact operators on C(K). If T is weakly compact, the associated vector measure μT is σ‑additive and enjoys a uniform continuity property that forces the images of any increasing chain of functions to become arbitrarily close, thus verifying (bishop). Conversely, assuming (bishop) for μT, the authors prove a topological version of Rosenthal’s lemma: for any ε > 0 there exists a family of pairwise disjoint clopen sets whose μT‑measure is smaller than ε. This lemma yields the σ‑additivity needed to deduce that T is weakly compact.

A second major contribution is the structural analysis of the class 𝔅(K) of operators on C(K) satisfying (bishop). Using a Ramsey‑type theorem of Dushnik and Miller, the authors show that 𝔅(K) is closed under left multiplication by arbitrary bounded operators on C(K) and under norm limits; consequently 𝔅(K) forms a closed left ideal in the Banach algebra B(C(K)). The proof hinges on the ability to thin out any chain of functions into a subchain on which the left multiplier behaves uniformly, thereby preserving the Cauchy property required by (bishop).

The paper also provides concrete examples of compact spaces K for which the identity operator Id : C(K) → C(K) satisfies (bishop). Two families are highlighted:

• Locally connected compact spaces with countable cellularity. In such spaces one can cover K by a countable family of pairwise disjoint open sets; any increasing chain of continuous functions can be refined to a chain supported on a decreasing sequence of these sets, guaranteeing the Cauchy property for the identity.
• Ladder‑system spaces (constructed on ω₁ using a ladder system). Despite their non‑metrizable and highly disconnected nature, the ladder structure allows one to define chains whose supports shrink along the ladder, again forcing the identity to satisfy (bishop).

These examples demonstrate that (bishop) is not confined to “nice’’ spaces; it can hold even in pathological settings, provided the underlying topology admits a suitable combinatorial decomposition.

Finally, the authors discuss the relationship of (bishop) to existing concepts in Banach space theory. While weakly compact operators are known to be Dunford–Pettis and to map weakly null sequences to norm‑null sequences, operators satisfying (bishop) inherit a weaker version of this behavior: they map any increasing chain to a norm‑Cauchy chain. The ideal structure uncovered suggests that (bishop) could serve as a bridge between the classical weakly compact ideal and larger ideals such as the strictly singular or inessential operators. The paper ends with several open problems, notably whether (bishop) can be characterised for broader classes of spaces (e.g., non‑extremally disconnected K) and how the condition interacts with spectral theory and the geometry of Banach spaces.