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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Three-space problems for the approximation properties


Authors: Gilles Godefroy and Pierre David Saphar
Journal: Proc. Amer. Math. Soc. 105 (1989), 70-75
MSC: Primary 46B20
MathSciNet review: 930249
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Abstract: Let $ M$ be a closed subspace of a Banach space $ X$ . We suppose that $ M$ has the B.A.P. and that $ {M^ \bot }$ is complemented in $ {X^*}$. Then, if $ X/M$ has the B.A.P. (resp. the A.P.), the space $ X$ has the same property. There are similar results if $ M$ is an $ {\mathcal{L}_\infty }$ space. If $ X/M$ is an $ {\mathcal{L}_1}$ space, then $ X$ has the B.A.P. if and only if $ M$ has the B.A.P. We notice that the quotient algebra $ L(H)/K(H)$ ( $ H$ infinite-dimensional Hilbert space) does not have the A.P.


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DOI: https://doi.org/10.1090/S0002-9939-1989-0930249-6
Keywords: approximation property, three-space problem, extension of finite rank operators
Article copyright: © Copyright 1989 American Mathematical Society