Algebraic and topological reflexivity properties of lp(X) spaces

We show that the isometry group of lp(X), for a separable Banach space X, is algebraically reflexive if and only if the isometry group of X is algebraically reflexive. We also show that the topological reflexivity of the isometry group of X does not imply that of lp(X). Moreover, we give a generalization of these results for a class of substitution spaces PX{Bn}.