Lie theoretic approach to unitary groups of 𝐶*-algebras

Abstract

Following Robert’s [J. Reine Angew. Math. 756 (2019), pp. 285–319], we study the structure of unitary groups and groups of approximately inner automorphisms of unital C ∗ C^* -algebras, taking advantage of the former being Banach-Lie groups. For a given unital C ∗ C^* -algebra A A , we provide a description of the closed normal subgroup structure of the connected component of the identity of the unitary group, denoted by U A U_A , resp. of the subgroup of approximately inner automorphisms induced by the connected component of the identity of the unitary group, denoted by V A V_A , in terms of perfect ideals, i.e. ideals admitting no characters. When the unital algebra is locally AF, we show that there is a one-to-one correspondence between closed normal subgroups of V A V_A and perfect ideals of the algebra, which can be in the separable case conveniently described using Bratteli diagrams; in particular showing that every closed normal subgroup of V A V_A is perfect. We also characterize unital C ∗ C^* -algebras A A such that U A U_A , resp. V A V_A are topologically simple, generalizing the main results of Robert [J. Reine Angew. Math. 756 (2019), pp. 285–319] from \cite{Rob19}. In the other way round, under certain conditions, we characterize simplicity of the algebra in terms of the structure of the unitary group. This in particular applies to reduced group C ∗ C^* -algebras of discrete groups and we show that when A A is a reduced group C ∗ C^* -algebra of a non-amenable countable discrete group, then A A is simple if and only if U A / T U_A/\mathbb {T} is topologically simple.