Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-14T21:40:15.940Z Has data issue: false hasContentIssue false
  • English
  • Français

Left approximate identities in algebras of compact operators on Banach spaces

Published online by Cambridge University Press:  14 November 2011

P. G. Dixon
P. G. Dixon
Affiliation:
Department of Pure Mathematics, University of Sheffield, Sheffield S3 7RH, England
Get access Rights & Permissions [Opens in a new window]

Synopsis

We study the existence of left approximate units, left approximate identities and bounded left approximate identities in the algebras (X)of all compact operators on a Banach space X and ℱ(X)− of all operators uniformly approximable by finite rank operators. In the case of bounded left approximate identities, necessary and sufficient conditions on X are obtained. In the other cases, sufficient conditions are obtained, together with an example of non-existence using a space constructed by Szankowski. The possibility of the sufficient conditions being also necessary depends on the question of whether every compact set is contained in the closure of the image of the unit ball under an operator in (X)(or ℱ(X)−). Sufficient conditions on X are obtained for this to be true, but it is conjectured that the answer for general X is negative.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 104 , Issue 1-2 , 1986 , pp. 169 - 175
Copyright
Copyright © Royal Society of Edinburgh 1986

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Altman, M.. Contracteurs dans les algebres de Banach. C.R. Acad. Sci. Paris Sér. A-B 274 (1972), 399–400.Google Scholar
2Bonsall, F. F. and Duncan, J.. Complete normed algebras (Berlin: Springer, 1973).CrossRefGoogle Scholar
3Doran, R. S. and Wichmann, J.. Approximate identities and factorization in Banach modules. Lecture Notes in Mathematics 768 (Berlin: Springer, 1979).Google Scholar
4Lindenstrauss, J. and Tzafriri, L.. Classical Banach spaces I: sequence spaces (Berlin: Springer, 1977).CrossRefGoogle Scholar
5Lindenstrauss, J. and Tzafriri, L.. Classical Banach spaces II: function spaces (Berlin: Springer, 1979).CrossRefGoogle Scholar
6Szankowski, A.. Subspaces without the approximation property. Israel J. Math. 30 (1978), 123–129.CrossRefGoogle Scholar