Nonweakly compact operators from order-Cauchy complete $C(S)$ lattices, with application to Baire classes
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- by Frederick K. Dashiell PDF
- Trans. Amer. Math. Soc. 266 (1981), 397-413 Request permission
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.References
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Additional Information
- © Copyright 1981 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 266 (1981), 397-413
- MSC: Primary 47B55; Secondary 26A21, 28A60, 46E05
- DOI: https://doi.org/10.1090/S0002-9947-1981-0617541-7
- MathSciNet review: 617541