Hostname: page-component-848d4c4894-sjtt6 Total loading time: 0 Render date: 2024-06-30T18:16:45.622Z Has data issue: false hasContentIssue false
  • English
  • Français

M-Ideals in L(l1, E)

Published online by Cambridge University Press:  20 November 2018

D. J. Fleming  and
D. M. Giarrusso
D. J. Fleming
Affiliation:
Department of Mathematics, St. Lawrence UniversityCanton, NY 13617, U.S.A.
D. M. Giarrusso
Affiliation:
Department of Mathematics, St. Lawrence UniversityCanton, NY 13617, U.S.A.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this article it is shown that for any Banach space E,L (l1,E) always contains uncountably many distinct A/-ideals that are closed subspaces of K(l1,E) and which are not complemented in L (l1,E) . Using standard duality arguments one obtains the result that infinitely many distinct subspaces of K(E, c0) are M-ideals in L(E, c0). In particular, for the case E = c0, this shows that the uniqueness conditions enjoyed by K(lp), p > 1, is not valid for E = c0. The results are obtained by utilizing the identification of L (l1,E) with the vector-valued sequence space lx(E) and to exploit natural decompositions of lx(E)’ afforded by a class of Lprojections on lx(E)’ induced by certain E'-valued vector measures.

Keywords

Primary 46B20Secondary 46A32
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 29 , Issue 1 , 01 March 1986 , pp. 3 - 10
Copyright
Copyright © Canadian Mathematical Society 1986

References

1. Alfsen, E. M. and Effros, E. G., Structure in real Banach spaces I, Ann. of Math., 96 (1972), pp. 98–173.Google Scholar
2. Behrends, E., et al., U structure in real Banach spaces, 613, Springer Lecture Notes.Google Scholar
3. Behrends, E., “M structure and the Banach Stone Theorem ” 736, Springer Lecture Notes.Google Scholar
4. Cunningham, F., L-structures in L-spaces, Trans. Amer. Math. Soc, 95 (1960), pp. 274—299.Google Scholar
5. Dixmier, J, Les fonctionnelles linéaires sur l’ ensembles des opérateurs bornés d'un espace de Hubert, Ann. of Math, 51 (1950), pp. 387–408.Google Scholar
6. Fleming, D. J. and Giarrusso, D. M., Topological decompositions of the duals of locally convex operator spaces, Math. Proc. Camb. Phil. Soc, 93 (1983), pp. 307–314.Google Scholar
7. Flynn, P., A characterization of M-ideals in B(lp)for J < p < ∞, pac . J. Math, 98 (1982), pp. 73–80.Google Scholar
8. Hennefeld, J. A., A decomposition for B(X)* and unique Hahn—Banach extensions, Pac. J. Math, 46 (1973), pp. 197–199.Google Scholar
9. Mach, J. and Ward, J., Approximation by compact operators on certain Banach spaces, J. Approximation Theory, 23 (1978), pp. 274–286.Google Scholar
10. Saatkamp, K., M-ideals of compact operators, Math. Z., 158 (1978), pp. 253–263.Google Scholar