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(TU), and ker(TU∗) 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.
