Invariant functionals and the uniqueness of invariant norms (Q1888366)

Let \((X, \| . \| )\) be a Banach space and let \(\tau\) be a representation of a compact group \(G\) on \(X\). The author studies whether \(X\) carries a unique invariant norm in the sense that \(\| . \| \) is the unique norm on \(X\) for which \(\tau\) is a representation. The approach relies on invariant functionals on \(X\). The uniqueness of the norm is characterized in terms of the automatic continuity of the invariant functionals in the case where \(X\) is a dual Banach space and \(\tau\) is a \(\sigma(X, X_1)\)-continuous representation of \(G\) on \(X\) such that \(\tau(G)\) consists of \(\sigma(X, X_1)\)-continuous operators, where \(X_1\) is a linear subspace of \(X_*\). The usefulness of this characterization is illustrated by considering the Banach space \(L^p(\Omega)\), where \(\Omega\) is a locally compact Hausdorff space equipped with a positive Radon measure on which \(G\) acts as a group of continuous invertible measure-preserving transformations. It turns out that the property that the classical norms \(| | . | | _p\) are the unique norms on \(L^p(\Omega)\) which are well-behaved with respect to translations is equivalent to the transitivity of the action and the property that every translation invariant functional on \(L^p(G)\) is a multiple of the Haar integral. Thus the results of [\textit{J. Extremera, J. F. Mena} and \textit{A. R. Villena} [J. Funct. Anal. 197, 212--227 (2003; Zbl 1056.43008)] are generalized. Moreover, a result of \textit{K. Jarosz} [J. Funct. Anal. 174, 417--429 (2000; Zbl 0981.46006)] is generalized.