A Cartan type theorem for finite-dimensional algebras

Let A be a finite direct sum of full matrix algebras over the complex field. We prove that if F is a holomorphic map of the open spectral unit ball of A into itself such that F(0)=0 and F′(0)=I, the identity of A, then a and F(a) have always the same spectrum. As an application we obtain a new proof, purely function-theoretic, of the fact that a unital spectral isometry on a finite-dimensional semi-simple Banach algebra is a Jordan morphism.