Some duality results on bounded approximation properties of pairs (Q2847828)

Let \(X\) be a Banach space and \(Y\) a closed subspace of \(X\). Let \(\lambda \geq 1\). Following \textit{T. Figiel} et al. [Isr. J. Math. 183, 199--231 (2011; Zbl 1235.46027)] the pair \((X,Y)\) is said to have the \(\lambda\)-bounded approximation property (BAP) if there exists a net of finite rank operators \((S_\alpha)\) on \(X\) such that \(\|S_\alpha\| \leq \lambda\), \(S_\alpha \to I_X\) pointwise, and \(S_\alpha(Y) \subset Y\) for all \(\alpha\). Letting \(Y = \{0\}\) or \(Y = X\), the \(\lambda\)-BAP for the pair \((X,Y)\) is the classical \(\lambda\)-BAP for \(X\).NEWLINENEWLINEUsing the principle of local reflexivity, one can show that, if \(X^*\) has the \(\lambda\)-BAP, then \(X\) has the \(\lambda\)-BAP. The corresponding question for pairs is: If the pair \((X^*,Y^\perp)\) has the \(\lambda\)-BAP, does the pair \((X,Y)\) have the \(\lambda\)-BAP?NEWLINENEWLINEThe main result of the present paper gives a positive answer to the above question by showing that \((X^*, Y^\perp)\) has the \(\lambda\)-BAP if and only if there exists a net of finite rank operators \((S_\alpha)\) on \(X\) such that \(\|S_\alpha\| \leq \lambda\), \(S_\alpha \to I_X\) pointwise, \(S_\alpha^* \to I_{X^*}\) pointwise, and \(S_\alpha(Y) \subset Y\) (hence \(S_\alpha^*(Y^\perp) \subset Y^\perp\)) for all \(\alpha\).