On Multiplicity of Uniform Norms and Maximal Spectral Substructures in Commutative Banach Algebras

Abstract

Let $\mathcal{A}$ be a semisimple commutative Banach algebra. It is shown that either $\mathcal{A}$ has exactly one uniform norm or it admits uncountably many uniform norms. Further, it is shown that there always exists a largest closed subalgebra of $\mathcal{A}$ which is weakly regular, and that there always exist largest closed ideals in $\mathcal{A}$ having unique uniform norm property (UUNP) and spectral extension property (SEP) respectively.