Invariant functionals and the uniqueness of invariant norms

Let τ be a representation of a compact group G on a Banach space (X,||·||). The question we address is whether X carries a unique invariant norm in the sense that ||·|| is the unique norm on X for which τ 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 τ is a -continuous representation of G on X such that τ(G) consists of -continuous operators. We illustrate the usefulness of this characterization by studying the uniqueness of the norm on the spaces Lp(Ω), where Ω is a locally compact Hausdorff space equipped with a positive Radon measure and G acts on Ω as a group of continuous invertible measure-preserving transformations.