Three-space problems for the approximation properties

Godefroy, Gilles; Saphar, Pierre David

doi:10.1090/S0002-9939-1989-0930249-6

Three-space problems for the approximation properties
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by Gilles Godefroy and Pierre David Saphar PDF

Proc. Amer. Math. Soc. 105 (1989), 70-75 Request permission

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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