On the automorphisms of the spectral unit ball

Jérémie Rostand¹

¹ Département de mathématiques et statistique Université Laval Québec, Canada G1K 7P4

Studia Mathematica, Tome 155 (2003) no. 3, pp. 207-230

Résumé

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 ${\mit \Omega }$.

Détail

DOI : 10.4064/sm155-3-2

Keywords: mit omega spectral unit ball mathbb set times matrices spectral radius interested classifying automorphisms mit omega know enough consider normalized automorphisms mit omega automorphisms satisfying where identity map mathbb known normalized automorphisms conjugations every normalized automorphism conjugation locally neighborhood matrix distinct eigenvalues answer yes prove normalized automorphism mit omega conjugation almost everywhere mit omega