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

Abstract

This paper is concerned with the connection between weak compactness properties in the duals of certain Banach spaces of type C ( S ) C(S) and order properties in the vector lattice C ( S ) C(S) . The weak compactness property of principal interest here is the condition that every nonweakly compact operator from C ( S ) C(S) into a Banach space must restrict to an isomorphism on some copy of l ∞ {l^\infty } in C ( S ) C(S) . (This implies Grothendieck’s property that every w ∗ {w^ \ast } -convergent sequence in C ( S ) ∗ 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 σ \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.