Amenability of Banach algebras of compact operators |
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Authors: | N Grønbæk B E Johnson G A Willis |
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Institution: | 1. Matematisk Institut, Universitetsparken 5, DK-2100, K?benhavn ?, Denmark 2. Department of Mathematics and Statistics, University of Newcastle upon Tyne, NE1 7RU, Newcastle upon Tyne, England 3. Department of Mathematics, The University of Newcastle, 2308, New South Wales, Australia
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Abstract: | In this paper we study conditions on a Banach spaceX that ensure that the Banach algebraК(X) of compact operators is amenable. We give a symmetrized approximation property ofX which is proved to be such a condition. This property is satisfied by a wide range of Banach spaces including all the classical
spaces. We then investigate which constructions of new Banach spaces from old ones preserve the property of carrying amenable
algebras of compact operators. Roughly speaking, dual spaces, predual spaces and certain tensor products do inherit this property
and direct sums do not. For direct sums this question is closely related to factorization of linear operators. In the final
section we discuss some open questions, in particular, the converse problem of what properties ofX are implied by the amenability ofК(X).
BEJ supported by MSRVP at Australian National University; GAW supported by SERC grant GR-F-74332. |
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