Application on Absolutely Summing Operators on a Tree Space and the Bounded Approximation Property

For describing the space P (C [0, 1], X), where X is a Banach space, of absolutely summing operators from C [0, 1] to X in terms of the space X itself, we construct a tree space l_1^tree (X) on X. It consists of special trees in X which we call two-trunk trees. We show that P (C [0, 1], X) is isometrically isomorphic to l_1^tree (X). As an application on [19], we characterize the bounded approximation property (BAP) and the weak BAP in terms of X^*-valued square sequence spaces.