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