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Abstract

Let ${\mit \Omega }$ be the spectral unit ball of $M_n({\mathbb C})$, that is, the set of $n\times n$ matrices with spectral radius less than 1. We are interested in classifying the automorphisms of ${\mit \Omega }$. We know that it is enough to consider the normalized automorphisms of ${\mit \Omega }$, that is, the automorphisms $F$ satisfying $F(0)=0$ and $F'(0)=I$, where $I$ is the identity map on $M_n({\mathbb C})$. The known normalized automorphisms are conjugations. Is every normalized automorphism a conjugation? We show that locally, in a neighborhood of a matrix with distinct eigenvalues, the answer is yes. We also prove that a normalized automorphism of ${\mit \Omega }$ is a conjugation almost everywhere on ${\mit \Omega }$.