On uniform topological algebras

TL;DR

This paper investigates the uniqueness of uniform norms in uniform topological algebras, establishing conditions under which the uniform norm is the only such norm and exploring properties of spectra and algebra norms.

Contribution

It proves the uniqueness of the uniform norm in certain classes of uniform topological algebras and characterizes algebra norms in regular semisimple commutative Banach algebras.

Findings

The uniform norm is the only uniform norm on a uniform normed Q-algebra.

In regular uniform normed Q-algebras with a unit, the uniform norm is unique.

A uniform topological algebra with an equicontinuous spectrum is a uniform normed algebra.

Abstract

The uniform norm on a uniform normed Q-algebra is the only uniform Q-algebra norm on it. The uniform norm on a regular uniform normed Q-algebra with unit is the only uniform norm on it. Let A be a uniform topological algebra whose spectrum M (A) is equicontinuous, then A is a uniform normed algebra. Let A be a regular semisimple commutative Banach algebra, then every algebra norm on A is a Q-algebra norm on A.