A Cartan theorem for Banach algebras

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Let A A be a semisimple Banach algebra, and let Ω A \Omega _A be its spectral unit ball. We show that every holomorphic map G : Ω A → Ω A G\colon \Omega _A\to \Omega _A satisfying G ( 0 ) = 0 G(0)=0 and G ′ ( 0 ) = I G’(0)=I fixes those elements of Ω A \Omega _A which belong to the centre of A A , but not necessarily any others. Using this, we deduce that the automorphisms of Ω A \Omega _A all leave the centre invariant. As a further application, we give a new proof of Nagasawa’s generalization of the Banach-Stone theorem.