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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:, gronbaek@math.ku.dk
George A. Willis
Affiliation:
Centre of Mathematical Analysis Australian National University GPO Box 4 Canberra, ACT 2601 Australia
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Abstract

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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

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