Commuting holomorphic maps on the spectral unit ball

We prove that if F is a holomorphic map from the open spectral unit ball of a primitive Banach algebra into itself satisfying F(0) = 0, F(') (0) = I and F(x) x = xF(x) for every x, then F is the identity map. Using this, we prove that if is a semisimple Banach algebra and is a primitive Banach algebra, then any unital spectral isometry from onto which locally preserves commutativity is a Jordan morphism. The same is true when and are both assumed to be von Neumann algebras.