Lifting Convex Approximation Properties and Cyclic Operators with Vector Lattices

Abstract

We demonstrate 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. We show that certain adjoint multiplication operators are convex- cyclic and show that some are convex- cyclic but no convex polynomial of the operator is hypercyclic. Also some adjoint multi-plication operators are convex- cyclic but not 1-weakly hypercyclic. We deal with two weaker forms of injectivity which turn out to have a rich structure behind: separable injectivity and universal separable injectivity. We show several structural and stability properties of these classes of Banach spaces. We provide natural examples of separably injective spaces, including L_∞ ultraproducts built over countably incomplete ultrafilters, in spite of the fact that these ultraproducts are never injective. We show that the Fremlin tensor product is not square mean complete when the two spaces are uncountable metrizable compact spaces.