Remote Access Transactions of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9947-1981-0617541-7
MathSciNet review: 617541
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] N. Aronszajn and P. Panitchpakdi, Extensions of uniformly continuous transformations and hyperconvex metric spaces, Pacific J. Math. 6 (1956), 405-439. MR 0084762 (18:917c)
  • [2] F. Dashiell, Isomorphism problems for the Baire classes, Pacific J. Math. 52 (1974), 29-43. MR 0355563 (50:8037)
  • [3] -, Weakly unconditionally summable sequences of continuous functions (preprint).
  • [4] F. Dashiell, A. Hager and M. Henriksen, Order-Cauchy completions of rings and vector lattices of continuous functions, Canad. J. Math. 32 (1980), 657-685. MR 586984 (81k:46020)
  • [5] D. H. Fremlin, Topological Riesz spaces and measure theory, Cambridge Univ. Press, Cambridge, 1974. MR 0454575 (56:12824)
  • [6] Z. Frolik, Three uniformities associated with uniformly continuous functions, Proc. Rome Conf. on Rings of Continuous Functions, 1973, Symposia Mathematica, vol. 17, Academic Press, New York, 1976, pp. 69-80. MR 0478110 (57:17599)
  • [7] L. Gillman and M. Jerison, Rings of continuous functions, Van Nostrand Reinhold, Princeton, N. J., 1960. MR 0116199 (22:6994)
  • [8] A. Grothendieck, Topological vector spaces, translated by O. Chaljub, Gordon and Breach, New York, 1973. MR 0372565 (51:8772)
  • [9] A. W. Hager, Real-valued functions on Alexandroff (zero-set) spaces, Comment. Math. Univ. Carolinae 16 (1975), 755-769. MR 0394547 (52:15348)
  • [10] H. Hahn, Reelle Funktionen, Chelsea, New York, 1948.
  • [11] F. Hausdorff, Set theory, 2nd ed., Chelsea, New York, 1962. MR 0141601 (25:4999)
  • [12] M. Henriksen and D. Johnson, On the structure of a class of Archimedean lattice-ordered algebras, Fund. Math. 50 (1961), 73-94. MR 0133698 (24:A3524)
  • [13] K. Kuratowski and M. Mostowski, Set theory, 2nd ed., North-Holland, Amsterdam, New York and Oxford, 1976.
  • [14] R. D. Mauldin, Baire functions, Borel sets, and ordinary function system, Advances in Math. 12 (1974), 418-450. MR 0367911 (51:4153)
  • [15] H. P. Rosenthal, On relatively disjoint families of measures, with some applications to Banach space theory, Studia Math. 37 (1970), 13-36; correction, 311-313. MR 0270122 (42:5015)
  • [16] H. L. Royden, Real analysis, 2nd ed., Macmillan, New York, 1968. MR 0151555 (27:1540)
  • [17] H. H. Schaefer, Banach lattices and positive operators, Springer-Verlag, New York, 1974. MR 0423039 (54:11023)
  • [18] G. Seever, Measures on $ F$ spaces, Trans. Amer. Math. Soc. 133 (1968), 267-280. MR 0226386 (37:1976)
  • [19] Z. Semadeni, Banach spaces of continuous functions, PWN, Warsaw, 1971.
  • [20] A. I. Veksler and V. A. Geiler, Order and disjoint completeness of linear partially ordered spaces, Sibirsk. Mat. Z. 13 (1972), 43-51 = Siberian Math. J. 13 (1972), 30-35. MR 0296654 (45:5713)
  • [21] C. J. Everett, Sequence completion of lattice moduls, Duke Math. J. 11 (1944), 109-119. MR 0009592 (5:169i)
  • [22] F. Papangelou, Order convergence and topological completion of commutative lattice-groups, Math. Ann. 155 (1964), 81-107. MR 0174498 (30:4699)
  • [23] J. Quinn, Intermediate Riesz spaces, Pacific J. Math. 56 (1975), 225-263. MR 0380355 (52:1255)
  • [24] A. I. Veksler, On one class of ordered sequentially continuous functions and regular Borel measures, Siberian Math. J. 17 (1976), 572-580. MR 0423041 (54:11025)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 47B55, 26A21, 28A60, 46E05

Retrieve articles in all journals with MSC: 47B55, 26A21, 28A60, 46E05


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1981-0617541-7
Article copyright: © Copyright 1981 American Mathematical Society

American Mathematical Society