Nonweakly compact operators from orderCauchy complete lattices, with application to Baire classes
Author:
Frederick K. Dashiell
Journal:
Trans. Amer. Math. Soc. 266 (1981), 397413
MSC:
Primary 47B55; Secondary 26A21, 28A60, 46E05
MathSciNet review:
617541
Fulltext PDF Free Access
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 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 orderCauchy 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.
 [1]
N.
Aronszajn and P.
Panitchpakdi, Extension of uniformly continuous transformations and
hyperconvex metric spaces, Pacific J. Math. 6 (1956),
405–439. MR 0084762
(18,917c)
 [2]
F.
K. Dashiell Jr., 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, OrderCauchy completions of rings and vector lattices of
continuous functions, Canad. J. Math. 32 (1980),
no. 3, 657–685. MR 586984
(81k:46020), http://dx.doi.org/10.4153/CJM19800520
 [5]
D.
H. Fremlin, Topological Riesz spaces and measure theory,
Cambridge University Press, London, 1974. MR 0454575
(56 #12824)
 [6]
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
(57 #17599)
 [7]
Leonard
Gillman and Meyer
Jerison, Rings of continuous functions, The University Series
in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton,
N.J.TorontoLondonNew York, 1960. MR 0116199
(22 #6994)
 [8]
A.
Grothendieck, Topological vector spaces, Gordon and Breach
Science Publishers, New York, 1973. Translated from the French by Orlando
Chaljub; Notes on Mathematics and its Applications. MR 0372565
(51 #8772)
 [9]
Anthony
W. Hager, Realvalued functions on Alexandroff (zeroset)
spaces, Comment. Math. Univ. Carolinae 16 (1975),
no. 4, 755–769. MR 0394547
(52 #15348)
 [10]
H. Hahn, Reelle Funktionen, Chelsea, New York, 1948.
 [11]
Felix
Hausdorff, Set theory, Second edition. Translated from the
German by John R. Aumann et al, Chelsea Publishing Co., New York, 1962. MR 0141601
(25 #4999)
 [12]
M.
Henriksen and D.
G. Johnson, On the structure of a class of archimedean
latticeordered algebras., Fund. Math. 50
(1961/1962), 73–94. MR 0133698
(24 #A3524)
 [13]
K. Kuratowski and M. Mostowski, Set theory, 2nd ed., NorthHolland, Amsterdam, New York and Oxford, 1976.
 [14]
R.
Daniel Mauldin, Baire functions, Borel sets, and ordinary function
systems, Advances in Math. 12 (1974), 418–450.
MR
0367911 (51 #4153)
 [15]
Haskell
P. Rosenthal, On relatively disjoint families of measures, with
some applications to Banach space theory, Studia Math.
37 (1970), 13–36. MR 0270122
(42 #5015)
 [16]
H.
L. Royden, Real analysis, The Macmillan Co., New York, 1963.
MR
0151555 (27 #1540)
 [17]
Helmut
H. Schaefer, Banach lattices and positive operators,
SpringerVerlag, New York, 1974. Die Grundlehren der mathematischen
Wissenschaften, Band 215. MR 0423039
(54 #11023)
 [18]
G.
L. Seever, Measures on 𝐹spaces,
Trans. Amer. Math. Soc. 133 (1968), 267–280. MR 0226386
(37 #1976), http://dx.doi.org/10.1090/S00029947196802263865
 [19]
Z. Semadeni, Banach spaces of continuous functions, PWN, Warsaw, 1971.
 [20]
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
(45 #5713)
 [21]
C.
J. Everett, Sequence completion of lattice moduls, Duke Math.
J. 11 (1944), 109–119. MR 0009592
(5,169i)
 [22]
Fredos
Papangelou, Order convergence and topological completion of
commutative latticegroups, Math. Ann. 155 (1964),
81–107. MR
0174498 (30 #4699)
 [23]
J.
Quinn, Intermediate Riesz spaces, Pacific J. Math.
56 (1975), no. 1, 225–263. MR 0380355
(52 #1255)
 [24]
A.
I. Veksler, A class of sequentially ordercontinuous functionals
and a class of regular Borel measures, Sibirsk. Mat. Ž.
17 (1976), no. 4, 757–767 (Russian). MR 0423041
(54 #11025)
 [1]
 N. Aronszajn and P. Panitchpakdi, Extensions of uniformly continuous transformations and hyperconvex metric spaces, Pacific J. Math. 6 (1956), 405439. MR 0084762 (18:917c)
 [2]
 F. Dashiell, Isomorphism problems for the Baire classes, Pacific J. Math. 52 (1974), 2943. MR 0355563 (50:8037)
 [3]
 , Weakly unconditionally summable sequences of continuous functions (preprint).
 [4]
 F. Dashiell, A. Hager and M. Henriksen, OrderCauchy completions of rings and vector lattices of continuous functions, Canad. J. Math. 32 (1980), 657685. 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. 6980. 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, Realvalued functions on Alexandroff (zeroset) spaces, Comment. Math. Univ. Carolinae 16 (1975), 755769. 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 latticeordered algebras, Fund. Math. 50 (1961), 7394. MR 0133698 (24:A3524)
 [13]
 K. Kuratowski and M. Mostowski, Set theory, 2nd ed., NorthHolland, Amsterdam, New York and Oxford, 1976.
 [14]
 R. D. Mauldin, Baire functions, Borel sets, and ordinary function system, Advances in Math. 12 (1974), 418450. 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), 1336; correction, 311313. 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, SpringerVerlag, New York, 1974. MR 0423039 (54:11023)
 [18]
 G. Seever, Measures on spaces, Trans. Amer. Math. Soc. 133 (1968), 267280. 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), 4351 = Siberian Math. J. 13 (1972), 3035. MR 0296654 (45:5713)
 [21]
 C. J. Everett, Sequence completion of lattice moduls, Duke Math. J. 11 (1944), 109119. MR 0009592 (5:169i)
 [22]
 F. Papangelou, Order convergence and topological completion of commutative latticegroups, Math. Ann. 155 (1964), 81107. MR 0174498 (30:4699)
 [23]
 J. Quinn, Intermediate Riesz spaces, Pacific J. Math. 56 (1975), 225263. 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), 572580. 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:
http://dx.doi.org/10.1090/S00029947198106175417
PII:
S 00029947(1981)06175417
Article copyright:
© Copyright 1981 American Mathematical Society
