Remark

Let $\widetilde{A}$ be the Banach algebra deduced from A by adjunction of a unit element $e$. Condition (ii) of prop. 1 shows that if $\widetilde{A}$ is regular, then A is regular. Suppose that A is regular and let us show that $\widetilde{A}$ is regular. Consider subsets F and $F'$ of $\mathsf{X}(\widetilde{A})$ which are disjoint and weakly closed (hence weakly compact) and construct an $x\in \widetilde{A}$ such that $\mathscr{G}(x)$ vanishes on F, and is equal to 1 on $F'$. Let $\chi_0\in \mathsf{X}(\widetilde{A})$ be the character which is zero on A. If $\chi_0\notin F'$, there exists, according to condition (iii) of prop. 1 and the assumption on A, an element $x\in A$ such that $\mathscr{G}(x)$ vanishes on F and is equal to 1 on $F'$. If $\chi_0\in F'$, we have $\chi_0\notin F$; there exists therefore an element $y\in A$ such that $\mathscr{G}(y)$ vanishes on $F'$ and is equal to 1 on F. The element $x=e-y$ of $\widetilde{A}$ then has the required property.