Invariant functionals and the uniqueness of invariant norms

Let tau be a representation of a compact group G on a Banach space (X, parallel to (.) parallel to). The question we address is whether X carries a unique invariant norm in the sense that parallel to (.) parallel to is the unique norm on X for which T is a representation. We characterize the uniqueness of norm in terms of the automatic continuity of the invariant functionals in the case when X is a dual Banach space and tau is a sigma(X, X-*)-continuous representation of G on X such that tau(G) consists of a sigma(X, X-*) -continuous operators. We illustrate the usefulness of this characterization by studying the uniqueness of the norm on the spaces L-p(Omega), where Omega is a locally compact Hausdorff space equipped with a positive Radon measure and G acts on Omega as a group of continuous invertible measure-preserving transformations. (C) 2003 Elsevier Inc. All rights reserved.