Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



When does $ {\rm ca}(\Sigma,X)$ contain a copy of $ l\sb \infty$ or $ c\sb 0$?

Author: Lech Drewnowski
Journal: Proc. Amer. Math. Soc. 109 (1990), 747-752
MSC: Primary 46E27; Secondary 46B20, 46G10
MathSciNet review: 1012927
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For a $ \sigma $-algebra $ \Sigma $ and a Banach space $ X,ca\left( {\Sigma ,X} \right)$ is the Banach space of all vector measures from $ \Sigma $ to $ X$. If $ \Sigma $ admits a nonzero atomless finite positive measure, then $ ca\left( {\Sigma ,X} \right) \supset {l_\infty }$ (or $ {c_0}$) if and only if there is a noncompact bounded linear operator from $ {l_2}$ to $ X$ (Theorem 1). Otherwise, $ ca\left( {\Sigma ,X} \right) \supset {l_\infty }$ (or $ {c_0}$) if and only if $ X \supset {l_\infty }$ (or $ {c_0}$) (Theorem 2).

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

  • [1] R. Anantharaman and J. Diestel, Sequences in the range of a vector measure, Comment. Math. Prace Mat. 30 (1991), no. 2, 221–235. MR 1122692
  • [2] Joseph Diestel, Sequences and series in Banach spaces, Graduate Texts in Mathematics, vol. 92, Springer-Verlag, New York, 1984. MR 737004
  • [3] J. Diestel and J. J. Uhl Jr., Vector measures, American Mathematical Society, Providence, R.I., 1977. With a foreword by B. J. Pettis; Mathematical Surveys, No. 15. MR 0453964
  • [4] Lech Drewnowski, Un théorème sur les opérateurs de 𝑙_{∞}(Γ), C. R. Acad. Sci. Paris Sér. A-B 281 (1975), no. 22, Aii, A967–A969 (French, with English summary). MR 0385626
  • [5] -, Copies of $ {l_\infty}$ in the operator space $ {K_{{w^ * }}}\left( {{X^*},Y} \right)$ (to appear).
  • [6] N. Dunford and J. T. Schwartz, Linear operators. Part I. Interscience, New York, 1958.
  • [7] Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. I, Springer-Verlag, Berlin-New York, 1977. Sequence spaces; Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 92. MR 0500056
  • [8] Haskell P. Rosenthal, On relatively disjoint families of measures, with some applications to Banach space theory, Studia Math. 37 (1970), 13–36. MR 0270122

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 46E27, 46B20, 46G10

Retrieve articles in all journals with MSC: 46E27, 46B20, 46G10

Additional Information

Keywords: Banach space, vector measure, copy of $ {l_\infty }$, copy of $ {c_0}$, Hilbert space $ {l_2}$, noncompact bounded linear operator
Article copyright: © Copyright 1990 American Mathematical Society