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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Nonweakly compact operators from order-Cauchy complete $ C(S)$ lattices, with application to Baire classes


Author: Frederick K. Dashiell
Journal: Trans. Amer. Math. Soc. 266 (1981), 397-413
MSC: Primary 47B55; Secondary 26A21, 28A60, 46E05
MathSciNet review: 617541
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Abstract: This paper is concerned with the connection between weak compactness properties in the duals of certain Banach spaces of type $ C(S)$ and order properties in the vector lattice $ C(S)$. The weak compactness property of principal interest here is the condition that every nonweakly compact operator from $ C(S)$ into a Banach space must restrict to an isomorphism on some copy of $ {l^\infty }$ in $ C(S)$. (This implies Grothendieck's property that every $ {w^ \ast }$-convergent sequence in $ C{(S)^ \ast }$ is weakly convergent.) The related vector lattice property studied here is order-Cauchy completeness, a weak type of completeness property weaker than $ \sigma $-completeness and weaker than the interposition property of Seever. An apphcation of our results is a proof that all Baire classes (of fixed order) of bounded functions generated by a vector lattice of functions are Banach spaces satisfying Grothendieck's property. Another application extends previous results on weak convergence of sequences of finitely additive measures defined on certain fields of sets.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1981-0617541-7
PII: S 0002-9947(1981)0617541-7
Article copyright: © Copyright 1981 American Mathematical Society