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We introduce a chain condition, defined for operators acting on  C ( K )-spaces, which is intermediate between weak compactness and having weakly compactly generated range. It is motivated by Pełczyński's characterization of weakly compact operators on  C ( K )-spaces. We prove that if  K  is extremally disconnected and  X  is a Banach space, then, for an operator  T  :  C ( K )→ X ,  T  is weakly compact if and only if  T  satisfiesif 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, 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 satisfyingforms a closed left ideal of  ℬ ( C ( K )).

A CHAIN CONDITION FOR OPERATORS FROM C ( K )-SPACES