Abstract

We introduce a notion of finite representability of dual Banach spaces in their subspaces preserving duality (f.d.-r in short) which arises in a natural way in situations such as the principle of local reflexivity. We give a characterization for the f.d.-r. which yields a version of the principle of local reflexivity, and can be applied to the study of the duality theory for ultrapowers of operators. For example, we show that the kernel ker(T^**_U) of an ultrapower of the second conjugate of an operator T is finitely representable in ker(T_U), and ker(T_U^*) is f.d.-r. in ker(T^*_U). Moreover, we study the duality properties of three semigroups of operators related with the superreflexivity and the finite representability of c₀ and ℓ₁ in a Banach space.

Authors