Regular norm and the operator semi-norm on a non-unital Banach Algebra

TL;DR

This paper characterizes when the regular norm on a commutative non-unital Banach algebra coincides with the operator semi-norm on its unitization, establishing a norm equivalence and algebraic structure condition.

Contribution

It provides a necessary and sufficient condition linking the regular norm and the operator semi-norm on the unitization of a commutative non-unital Banach algebra.

Findings

Regular norm equivalence with operator semi-norm on unitization

Characterization of Banach algebra structure via norms

Condition for the unitized algebra to be a Banach algebra

Abstract

We show that if $\mathfrak{A}$ is a commutative complex non-unital Banach Algebra with norm $\|\cdot\|$, then $\|\cdot\|$ is regular on $\mathfrak{A}$ if and only if $\|\cdot\|_{op}$ is a norm on $\mathfrak{A}\oplus \mathbb{C}$ and $\mathfrak{A}\oplus\mathbb{C}$ is a commutative complex Banach Algebra with respect to $\|\cdot\|_{op}$.