A note on spectrum preserving additive maps between $C^{*}$-algebras

Let $A$ be a unital $C^{*}$-algebra of real rank zero and $B$ a unital semi-simple Banach algebra. Let $\phi$ be a surjective spectrum preserving additive map from $A$ onto $B$. Then $\phi$ is continuous and Jordan isomorphism on self-adjoint element of $A$.