Absolutely summing operators on separable Lindenstrauss spaces as tree spaces and the bounded approximation property

Abstract

Let $X$ be a Banach space and let $Y$ be a separable Lindenstrauss space. We describe the Banach space $\mathcal{P}(Y,X)$ of absolutely summing operators as a general $\ell_1$-tree space. We also characterize the bounded approximation property and its weak version for $X$ in terms of the space of integral operators $\mathcal{I}(X,Z^*)$ and the space of nuclear operators $\mathcal{N}(X,Z^*)$, respectively, where $Z$ is a Lindenstrauss space, whose dual $Z^*$ fails to have the Radon-Nikodým property.