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

作者：Hart Klaas Pieter; Kania Tomasz^*; Kochanek Tomasz

来源：The Quarterly Journal of Mathematics, 2014, 65(2): 703-715.

DOI：10.1093/qmath/hat006

We introduce a chain condition %26lt;inline-graphic xlink:href=%26quot;HAT006IM1%26quot; xlink:type=%26quot;simple%26quot;/%26gt;, 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%26lt;remove%26gt;czyA%26quot;ski%26apos;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)-%26gt; X, T is weakly compact if and only if T satisfies %26lt;inline-graphic xlink:href=%26quot;HAT006IM2%26quot; xlink:type=%26quot;simple%26quot;/%26gt; 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%26apos;s lemma. We exhibit several compact Hausdorff spaces K for which the identity operator on C(K) satisfies %26lt;inline-graphic xlink:href=%26quot;HAT006IM3%26quot; xlink:type=%26quot;simple%26quot;/%26gt;, 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 %26lt;inline-graphic xlink:href=%26quot;HAT006IM4%26quot; xlink:type=%26quot;simple%26quot;/%26gt; forms a closed left ideal of a%26quot;not sign(C(K)).

• 出版日期2014-6