Holomorphic automorphisms and collective compactness in J*-algebras of operator

Abstract

Let G be the Banach-Lie group of all holomorphic automorphisms of the open unit ball \(B_\mathfrak{A} \) in a J*-algebra \(\mathfrak{A}\) of operators. Let \(\mathfrak{F}\) be the family of all collectively compact subsets W contained in \(B_\mathfrak{A} \). We show that the subgroup F ⊂ G of all those g ∈ G that preserve the family \(\mathfrak{F}\) is a closed Lie subgroup of G and characterize its Banach-Lie algebra. We make a detailed study of F when \(\mathfrak{A}\) is a Cartan factor.