The bidual of the compact operators

Palmer, Theodore W.

doi:10.1090/S0002-9947-1985-0776407-6

The bidual of the compact operators
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by Theodore W. Palmer PDF

Trans. Amer. Math. Soc. 288 (1985), 827-839 Request permission

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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