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

Dashiell, Frederick K.

doi:10.1090/S0002-9947-1981-0617541-7

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

N. Aronszajn and P. Panitchpakdi, Extension of uniformly continuous transformations and hyperconvex metric spaces, Pacific J. Math. 6 (1956), 405–439. MR 84762, DOI 10.2140/pjm.1956.6.405
F. K. Dashiell Jr., Isomorphism problems for the Baire classes, Pacific J. Math. 52 (1974), 29–43. MR 355563, DOI 10.2140/pjm.1974.52.29

Weakly unconditionally summable sequences of continuous functions

F. Dashiell, A. Hager, and M. Henriksen, Order-Cauchy completions of rings and vector lattices of continuous functions, Canadian J. Math. 32 (1980), no. 3, 657–685. MR 586984, DOI 10.4153/CJM-1980-052-0
D. H. Fremlin, Topological Riesz spaces and measure theory, Cambridge University Press, London-New York, 1974. MR 0454575, DOI 10.1017/CBO9780511897207
Zdeněk Frolík, Three uniformities associated with uniformly continuous functions, Symposia Mathematica, Vol. XVII (Convegno sugli Anelli di Funzioni Continue, INDAM, Rome, 1973) Academic Press, London, 1976, pp. 69–80. MR 0478110
Leonard Gillman and Meyer Jerison, Rings of continuous functions, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR 0116199, DOI 10.1007/978-1-4615-7819-2
A. Grothendieck, Topological vector spaces, Notes on Mathematics and its Applications, Gordon and Breach Science Publishers, New York-London-Paris, 1973. Translated from the French by Orlando Chaljub. MR 0372565
Anthony W. Hager, Real-valued functions on Alexandroff (zero-set) spaces, Comment. Math. Univ. Carolinae 16 (1975), no. 4, 755–769. MR 394547

Reelle Funktionen

Felix Hausdorff, Set theory, 2nd ed., Chelsea Publishing Co., New York, 1962. Translated from the German by John R. Aumann et al. MR 0141601
M. Henriksen and D. G. Johnson, On the structure of a class of archimedean lattice-ordered algebras, Fund. Math. 50 (1961/62), 73–94. MR 133698, DOI 10.4064/fm-50-1-73-94

Set theory

R. Daniel Mauldin, Baire functions, Borel sets, and ordinary function systems, Advances in Math. 12 (1974), 418–450. MR 367911, DOI 10.1016/S0001-8708(74)80011-3
Haskell P. Rosenthal, On relatively disjoint families of measures, with some applications to Banach space theory, Studia Math. 37 (1970), 13–36. MR 270122, DOI 10.4064/sm-37-1-13-36
H. L. Royden, Real analysis, The Macmillan Company, New York; Collier Macmillan Ltd., London, 1963. MR 0151555
Helmut H. Schaefer, Banach lattices and positive operators, Die Grundlehren der mathematischen Wissenschaften, Band 215, Springer-Verlag, New York-Heidelberg, 1974. MR 0423039, DOI 10.1007/978-3-642-65970-6
G. L. Seever, Measures on $F$-spaces, Trans. Amer. Math. Soc. 133 (1968), 267–280. MR 226386, DOI 10.1090/S0002-9947-1968-0226386-5

Banach spaces of continuous functions

A. I. Veksler and V. A. Geĭler, Order completeness and disjoint completeness of linear partially ordered spaces, Sibirsk. Mat. Ž. 13 (1972), 43–51 (Russian). MR 0296654
C. J. Everett, Sequence completion of lattice moduls, Duke Math. J. 11 (1944), 109–119. MR 9592, DOI 10.1215/S0012-7094-44-01112-9
Fredos Papangelou, Order convergence and topological completion of commutative lattice-groups, Math. Ann. 155 (1964), 81–107. MR 174498, DOI 10.1007/BF01344076
J. Quinn, Intermediate Riesz spaces, Pacific J. Math. 56 (1975), no. 1, 225–263. MR 380355, DOI 10.2140/pjm.1975.56.225
A. I. Veksler, A class of sequentially order-continuous functionals and a class of regular Borel measures, Sibirsk. Mat. Ž. 17 (1976), no. 4, 757–767 (Russian). MR 0423041