In *G. B. Folland - A Course in Abstract Harmonic Analysis* we can read the following

" **(1.15) Proposition.** Let be $A$ a (complex) commutative unital Banach algebra with unit $e$, let $x_0 \in A$ and suppose that one of the following holds:

(i) $A$ is generated by $x_0$ and $e$.

(ii) $x_0$ is invertible and $A$ is generated by $x_0$ and $x_0^{-1}$.

(iii) $A$ is a simmetric $*$-algebra and $A$ is generated by $x_0$, $x_0^*$ and $e$.

Then $\hat{x_0}$ (the Gelfand trasform of $x_0$) is a homeomorphism from the spectrum $\sigma(A)$ of $A$ to the spectrum $\sigma(x_0)$ of $x_0$. "

The proof is elementary and the main motivation to state it is to explain the relationship between the spectrum of the algebra and the spectrum of an element.

My question is: there exists a converse of Prop. 1.15? That is, if $A$ is a commutative unital Banach algebra with $x_0 \in A$ such that $\hat x_0 : \sigma(A) \to \sigma(x_0)$ is a homeomorphism, then is it true that one of (i),(ii),(iii) (and maybe some other similar cases) holds?

So I am asking about a classification of commutative unital Banach algebras whose spectrum is homeomorphism to the spectrum of an element.

Thank you in advance for any suggestion.