A Cartan type theorem for finite-dimensional algebras

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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.

AMS classification

46Hxx
32Hxx

Keywords

Spectrum
Holomorphic mappings
Spectral unit ball

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