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

Let X be a Banach space. For describing the space P(C[0, 1],X) of absolutely summing operators from C[0, 1] to X in terms of the space X itself, we construct a tree space tree 1 (X) on X. It consists of special trees in X which we call two-trunk trees. We prove that P(C[0, 1],X) is isometrically isomorphic to tree 1 (X). As an application, we characterize the bounded approximation property (BAP) and the weak BAP in terms of X∗-valued sequence spaces. © 2010 Elsevier Inc. All rights reserved