The range of a ring homomorphism from a commutative $C^{*}$-algebra

We prove that if a commutative semi-simple Banach algebra $\mathcal A$ is the range of a ring homomorphism from a commutative $C^{*}$-algebra, then $\mathcal A$ is $C^{*}$-equivalent, i.e. there are a commutative $C^{*}$-algebra $\mathcal B$ and a bicontinuous algebra isomorphism between $\mathcal A$ and $\mathcal B$. In particular, it is shown that the group algebras $L^{1}(\mathbb {R})$, $L^{1}(\mathbb {T})$ and the disc algebra $A(\mathbb {D})$ are not ring homomorphic images of $C^{*}$-algebras.