Lifting bounded approximation properties from Banach spaces to their dual spaces ✩

Based on a new reformulation of the bounded approximation property, we develop a unified approach to the lifting of bounded approximation properties from a Banach space X to its dual X∗. This encompasses cases when X has the unique extension property or X is extendably locally reflexive. In particular, it is shown that the unique extension property of X permits to lift the metric A-approximation property from X to X∗, for any operator ideal A, and that there exists a Banach space X such that X,X∗∗, . . . are extendably locally reflexive, but X∗,X∗∗∗, . . . are not. © 2005 Elsevier Inc. All rights reserved.