Abstract
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.
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