Bounded approximation properties in terms of \(C[0,1]\) (Q2879614)

Let BAP abbreviate bounded approximation property and \(\mathcal A\) be a Banach operator ideal. Recently, in [``Bounded approximation properties via integral and nuclear operators'', Proc. Am. Math. Soc. 138, No. 1, 287--297 (2010; Zbl 1193.46012)], the authors of the paper under review introduced the property ``\(\lambda\)-BAP for \(\mathcal A\)''. The \(\lambda\)-BAP for \(\mathcal A\) (we postpone definitions a little while) is designed so that:NEWLINENEWLINE(a) The \(\lambda\)-BAP for the bounded linear operators \(\mathcal L\) is nothing but the classical \(\lambda\)-BAP, andNEWLINENEWLINE(b) The weak \(\lambda\)-BAP is the \(\lambda\)-BAP for the weakly compact operators \(\mathcal W\).NEWLINENEWLINEAmong the main results from the above mentioned paper are:NEWLINENEWLINE(i) The \(\lambda\)-BAP for \(\mathcal L\) equals the \(\lambda\)-BAP for the integral operators \(\mathcal I\), and the \(\lambda\)-BAP for \(\mathcal I\) is determined by \(ba(\mathbb{N})\), the dual of \(\ell_\infty\).NEWLINENEWLINE(ii) The \(\lambda\)-BAP for \(\mathcal W\) equals the \(\lambda\)-BAP for the nuclear operators \(\mathcal N\), and the \(\lambda\)-BAP for \(\mathcal N\) is determined by \(\ell_1\).NEWLINENEWLINENow, the object of the paper under review is to show that \(\lambda\)-BAP for \(\mathcal I\) and \(\lambda\)-BAP for \(\mathcal N\) both are determined by the Banach space \(M[0,1]\), the dual of \(C[0,1]\). It is time for definitions: A Banach space \(X\) is said to have the \(\lambda\)-BAP for \(\mathcal A\) if for every Banach space \(Y\) and every \(T\in \mathcal{A}(X,Y)\) there exists a net (\(S_\alpha\)) of finite-rank operators on \(X\) such that \(S_\alpha\to I_X\) uniformly on compact subsets of \(X\) and NEWLINE\[NEWLINE \limsup_\alpha \|TS_\alpha\|_{\mathcal A}\leq \lambda \|T\|_{\mathcal A}.NEWLINE\]NEWLINE The \(\lambda\)-BAP for \(A\) is determined by \(Z\) if ``for every Banach space \(Y\)'' can be exchanged by ``for the Banach space \(Z\)''.NEWLINENEWLINESome words about the structure of proving the results: Section 2 of the paper contains the proof of the \(\mathcal{N}\)-case and the separable case of the \(\mathcal{I}\)-case. The hardest part is to lift to the non-separable \(\mathcal{I}\)-case, which is done in Section 3. This proof uses that a Banach space \(X\) has the \(\lambda\)-BAP if and only if every separable ideal of \(X\) has the \(\lambda\)-BAP and rests upon the passing to ideals of a certain approximation principle for spaces with the weak BAP (Theorem 3.2).