Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-20T17:36:50.005Z Has data issue: false hasContentIssue false
  • English
  • Français

Approximate Identities in Banach Algebras of Compact Operators

Published online by Cambridge University Press:  20 November 2018

Niels Grønbæk  and
George A. Willis
Niels Grønbæk
Affiliation:
Matematisk Institut Universitetsparken 5 DK-2100 København Ø, Denmark email:, [email protected]
George A. Willis
Affiliation:
Centre of Mathematical Analysis Australian National University GPO Box 4 Canberra, ACT 2601 Australia
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.

Let X be a Banach space and let A be a uniformly closed algebra of compact operators on X, containing the finite rank operators. We set up a general framework to discuss the equivalence between Banach space approximation properties and the existence of right approximate identities in A. The appropriate properties require approximation in the dual X* by operators which are adjoints of operators on X. We show that the existence of a bounded right approximate identity implies that of a bounded left approximate identity. We give examples to show that these properties are not equivalent, however. Finally, we discuss the well known result that, if X* has a basis, then X has a shrinking basis. We make some attempts to generalize this to various bounded approximation properties.

Keywords

46B2846H2047D3047D50
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 36 , Issue 1 , 01 March 1993 , pp. 45 - 53
Copyright
Copyright © Canadian Mathematical Society 1993

References

[B&P] Berkson, E. and Porta, H., Representations of B(X), J. Funct. Anal. 3(1969) 1–34.Google Scholar
[B&D]F. F Bonsall and Duncan, J., Complete Normed Algebras, Springer-Verlag, 1973, Berlin Heidelberg.Google Scholar
[Di] Diestel, J., Sequences and Series in Banach Spaces, Graduate Texts in Mathematics 92, Springer-Verlag, New York, 1984.Google Scholar
[D&U] Diestel, J. and Uhl, J. J., Vector Measures, Math. Surveys 15, Amer. Math. Soc, 1977.Google Scholar
[D] Dixon, P. G., Left approximate identities in algebras of compact operators on Banach spaces, Proc. Royal Soc. Edinburgh 104A(1989) 169–175.Google Scholar
[D&W] Doran, R. S. and Wichmann, J., Approximate identities and factorization in Banach modules, Lecture Notes in Mathematics 768, Springer-Verlag, 1979, Berlin Heidelberg.Google Scholar
[D&S] Dunford, N. and Schwartz, J. T., Linear Operators, Part 1, Interscience, New York, 1958.Google Scholar
[Gr] Grothendieck, A., Résumé de la théorie métrique des produits tensoriels topologiques, Bol. Soc. Mat. Säo Paulo 8(1953) 1–70.Google Scholar
[G J&W]N. Gr0nbaek, Johnson, B. E. and Willis, G. A., Amenability of Banach algebras of compact operators, forthcoming paper.Google Scholar
[J]B.Johnson, E., Cohomology in Banach algebras, Mem. Amer. Math. Soc. 127(1972).Google Scholar
[J,R&Z] Johnson, W. B., Rosenthal, H. P. and Zippin, M., On bases, finite-dimensional decompositions and weaker structures in Banach spaces, Israel J. Math. 9(1971) 488–506.Google Scholar
[L&T] Lindenstrauss, J. and Tzafriri, L., Classical Banach Spaces 1, Springer-Verlag, 1977, Berlin Heidelberg.Google Scholar
[Pie] Pietsch, A., Operator Ideals, North-Holland, Berlin, 1980.Google Scholar
[Pis] Pisier, G., Factorization of Linear Operators and Geometry of Banach Spaces, Regional Conference Series in Mathematics 60, Amer. Math. Soc, Providence, Rhode Island, 1986.Google Scholar
[S] Szankowski, A., B(H) does not have the approximation property, Acta. Math. 147(1981) 89–108.Google Scholar
[Sa] Samuel, C., Bounded approximate identities in the algebra of compact operators on a Banach space, Proc. Amer. Math. Soc, to appear.Google Scholar
[W] Willis, G. A., The compact approximation property does not imply the approximation property, forthcoming paper.Google Scholar