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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Banach spaces which are $ M$-ideals in their biduals


Authors: Peter Harmand and Åsvald Lima
Journal: Trans. Amer. Math. Soc. 283 (1984), 253-264
MSC: Primary 46B10; Secondary 47D30
MathSciNet review: 735420
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Abstract: We investigate Banach spaces $ X$ such that $ X$ is an $ M$-ideal in $ {X^{{\ast}{\ast}}}$. Subspaces, quotients and $ {c_0}$-sums of spaces which are $ M$-ideals in their biduals are again of this type. A nonreflexive space $ X$ which is an $ M$-ideal in $ {X^{{\ast}{\ast}}}$ contains a copy of $ {c_0}$. Recently Lima has shown that if $ K(X)$ is an $ M$-ideal in $ L(X)$ then $ X$ is an $ M$-ideal in $ {X^{{\ast}{\ast}}}$. Here we show that if $ X$ is reflexive and $ K(X)$ is an $ M$-ideal in $ L(X)$, then $ K{(X)^{{\ast}{\ast}}}$ is isometric to $ L(X)$, i.e. $ K(X)$ is an $ M$-ideal in its bidual. Moreover, for real such spaces, we show that $ K(X)$ contains a proper $ M$-ideal if and only if $ X$ or $ {X^{\ast}}$ contains a proper $ M$-ideal.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1984-0735420-4
PII: S 0002-9947(1984)0735420-4
Keywords: $ M$-ideal, bidual, $ M$-structure, spaces of compact operators
Article copyright: © Copyright 1984 American Mathematical Society