Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The bidual of the compact operators

Author: Theodore W. Palmer
Journal: Trans. Amer. Math. Soc. 288 (1985), 827-839
MSC: Primary 47D30; Secondary 46B20, 46M05, 47D15
MathSciNet review: 776407
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Abstract: Let $ X$ be a Banach space such that $ {X^\ast}$ has the Radon-Nikodým property. If $ {X^\ast}$ also has the approximation property, then the Banach algebra $ B({X^{ \ast \ast }})$ of all bounded linear operators on $ {X^{\ast\ast}}$ is isometrically isomorphic (as an algebra) to the double dual $ {B_K}{(X)^{ \ast \ast }}$ of the Banach algebra of compact operators on $ X$ when $ {B_K}{(X)^{ \ast \ast }}$ is provided with the first Arens product. The chief result of this paper is a converse to the above statement. The converse is formulated in a strong fashion and a number of other results, including a formula for the second Arens product, are also given.

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Keywords: Compact operators, Arens products, approximation property, Radon-Nikodým property, nuclear operators
Article copyright: © Copyright 1985 American Mathematical Society