Absolutely summing operators on \(C[0,1]\) as a tree space and the bounded approximation property (Q600965)

Let \(X\) be a Banach space. For describing the space \(\mathcal P(C[0, 1], X)\) of absolutely summing operators from \(C[0,1]\) to \(X\) in terms of the space \(X\) itself, the authors construct a tree space \(\ell _1^{\text{tree}}(X)\) on \(X\) which relies on linear \(B\)-splines. It consists of special trees in \(X\) which the authors call two-trunk trees. They prove that \(\mathcal P(C[0, 1], X)\) is isometrically isomorphic to \(\ell _1^{\text{tree}}(X)\). As an immediate application, the authors provide a new equivalent formulation of the Radon-Nikodým property of Banach spaces. Furthermore, they characterize the \(\lambda\)-bounded approximation property (\(\lambda\)-BAP) and the weak \(\lambda\)-BAP in terms of \(X^{*}\)-valued sequence spaces. This sheds some new light on the still open problem whether the weak \(\lambda\)-BAP is strictly weaker than the \(\lambda\)-BAP, and, consequently, on the long-standing famous open problem whether the AP of a dual Banach space implies the \(1\)-BAP.