Three-space problem for some approximation properties
Let \(M\) be a closed subspace of a Banach space \(X\). In the case where \(M^\perp\) is complemented in the dual \(X^*\), \textit{G.\,Godefroy} and \textit{P.\,D.\thinspace Saphar} [Proc.\ Am.\ Math.\ Soc.\ 105, No.\,1, 70--75 (1989; Zbl 0674.46009)] studied the three-space problem for the approximation property (AP) and the bounded approximation property (BAP). In particular, Godefroy and Saphar showed that if \(M\) and \(X/M\) have the BAP, then \(X\) has the BAP whenever \(M^\perp\) is complemented in \(X^*\). The author follows the lead of Godefroy and Saphar and proves similar results for the weak BAP (as defined by \textit{Å.\,Lima} and \textit{E.\,Oja} [Math.\ Ann.\ 333, No.\,3, 471--484 (2005; Zbl 1097.46012)]). It is also shown that there is a space \(X\) with the BAP and a subspace \(M\) with the BAP such that \(K(X/M)\) is not in the closure of \(F(X/M)\) in the topology of uniform convergence on compact sets, and in particular does not have the AP, even though \(M^\perp\) is complemented in \(X^*\).
scientific article; zbMATH DE number 5707741