Lifting convex approximation properties from Banach spaces to their dual spaces and the related local reflexivity

Our main result demonstrates that rather weak forms of the extendable local reflexivity and of the principle of local reflexivity are needed for the lifting of bounded convex approximation properties from Banach spaces to their dual spaces. This provides a unified approach to the lifting of various bounded approximation properties, including, besides the classical ones, the approximation property for pairs of Banach spaces, and the positive approximation property of Banach lattices.