  We prove that if $\mathcal{A}$ is a complex, unital semisimple Banach algebra
and $\mathcal{B}$ is a complex, unital Banach algebra having a separating
family of finite-dimensional irreducible representations, then any unital
linear operator from $\mathcal{A}$ onto $\mathcal{B}$ which preserves the
spectral radius is a Jordan morphism.
