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

TL;DR

This paper investigates the uniqueness and multiplicity of uniform norms in semisimple commutative Banach algebras, revealing conditions for their existence and structure of maximal subalgebras and ideals.

Contribution

It establishes that such algebras either have a unique uniform norm or uncountably many, and identifies largest subalgebras and ideals with specific norm properties.

Findings

Either exactly one uniform norm or uncountably many exist in the algebra.

Existence of a largest weakly regular subalgebra.

Existence of largest ideals with UUNP and SEP.

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.