The convex approximation property of Banach spaces

We study the convex approximation property of Banach spaces to provide a unified approach to various approximation properties including, besides the classical ones, e.g., the positive approximation property of Banach lattices and the approximation property for pairs of Banach spaces. Our main results concern lifting of metric and weak metric approximation properties from Banach spaces to their dual spaces. As an easy application, it follows that if X⁎ or X⁎⁎ has the Radon–Nikodým property, then the approximation property of X⁎, defined by a convex subset of conjugate compact operators containing 0 (in particular, the positive approximation property of X⁎), is metric.