Nonweakly compact operators from order-Cauchy complete 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

DOI:
https://doi.org/10.1090/S0002-9947-1981-0617541-7

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 and order properties in the vector lattice . The weak compactness property of principal interest here is the condition that every nonweakly compact operator from into a Banach space must restrict to an isomorphism on some copy of in . (This implies Grothendieck's property that every -convergent sequence in is weakly convergent.) The related vector lattice property studied here is *order-Cauchy completeness*, a weak type of completeness property weaker than -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:
https://doi.org/10.1090/S0002-9947-1981-0617541-7

Article copyright:
© Copyright 1981
American Mathematical Society