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

Abstract

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