Hostname: page-component-7479d7b7d-t6hkb Total loading time: 0 Render date: 2024-07-11T14:50:43.639Z Has data issue: false hasContentIssue false
  • English
  • Français

Some algebraic properties of F(X) and K(X)

Published online by Cambridge University Press:  20 January 2009

Freda E. Alexander
Freda E. Alexander
Affiliation:
University of Glasgow, Glasgow G12 8WQ
Rights & Permissions [Opens in a new window]

Extract

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.

Throughout we consider operators on a reflexive Banach space X. We consider certain algebraic properties of F(X), K(X) and B(X) with the general aim of examining their dependence on the possession by X of the approximation property. B(X) (resp. K(X)) denotes the algebra of all bounded (resp. compact) operators on X and F(X) denotes the closure in B(X) of its finite rank operators. The two questions we consider are:

(1) Is K(X) equal to the set of all operators in B(X) whose right and left multiplication operators on F(X) (or on B(X)) are weakly compact?

(2) Is F(X) a dual algebra?

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 19 , Issue 4 , September 1975 , pp. 353 - 361
Copyright
Copyright © Edinburgh Mathematical Society 1975

References

REFERENCES

(1) Alexander, F. E., Compact and finite rank operators on subspaces of lp, Bull. London Math. Soc. (to appear).Google Scholar
(2) Bonsall, F. F. and Goldie, A. W., Annihilator algebras, Proc. London Math. Soc. 4 (1954), 154–167.CrossRefGoogle Scholar
(3) Davie, A. M., A counter-example on dual Banach algebras, Bull. London Math. Soc. 5 (1973), 79–81.CrossRefGoogle Scholar
(4) Davie, A. M., The approximation problem for Banach spaces, Bull. London Math. Soc. 5 (1973), 261–266.CrossRefGoogle Scholar
(5) Dunford, N. and Schwartz, J., Linear operators, Vol. I (Interscience, New York, 1958).Google Scholar
(6) Gil De Lamadrid, J., Measures and tensors, Trans. Amer. Math. Soc. 114 (1965), 98–121.CrossRefGoogle Scholar
(7) Grothendieck, A., Produits tensorielles topologiques et espaces nucléaires (Memoirs of the American Math. Soc., No. 16 1955).Google Scholar
(8) Lindenstrauss, J. and Pelczynski, A., Absolutely summing operators in spaces and their applications, Studia Math. 29 (1968), 275–326.Google Scholar
(9) Marti, J. T., Introduction to the theory of bases (Springer-Verlag, Berlin, 1969).CrossRefGoogle Scholar
(10) Ogasawara, T., Finite dimensionality of certain Banach algebras, J. Sci. Hiroshima Univ. Ser.A 17 (1951), 539–363.Google Scholar
(11) Olubummo, A., Operators of finite rank in a reflexive Banach space, Pacific J. Math. 12 (1962), 1023–1027.Google Scholar
(12) Pelczynski, A., Projections in certain Banach spaces, Studia Math. 19 (1960), 209–228.CrossRefGoogle Scholar
(13) Schatten, R., A theory of cross-spaces (Ann. Of Math. Studies 26, Princeton, 1960).Google Scholar