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

14 March 2014, 13:30–15:00
Fields institute, Room 210

Speaker: Tomasz Kania

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

Abstract:

Building upon work of Pełczyński, we introduce a chain condition, defined for operators acting on C(K)-spaces, which is weaker than weak compactness. We prove that if K is extremely disconnected and X is a Banach space then an operator T : C(K) -> X is weakly compact if and only if it satisfies our condition and this is if and only if the representing vector measure of T satisfies an analogous chain condition on Borel sets of K. As a tool for proving the above-mentioned result, we derive a topological counterpart of Rosenthal’s lemma. We exhibit classes of compact Hausdorff spaces K for which the identity operator on C(K) satisfies our condition, for instance every class of compact spaces that is preserved when taking closed subspaces and Hausdorff quotients, and which contains no non-metrisable linearly ordered space (like the classes of Eberlein spaces, Corson compact spaces etc.) serves as an example. Using a Ramsey-type theorem, due to Dushnik and Miller, we prove that the collection of operators on a C(K)-space satisfying our condition forms a closed left ideal of B(C(K)), however in general, it does not form a right ideal. This work is based on two papers (one joint with K. P. Hart and T. Kochanek and the second one joint with. R. Smith).

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