Lie theoretic approach to unitary groups of $C^*$-algebras

Abstract

Following Robert's [26], we study the structure of unitary groups and groups of approximately inner automorphisms of unital $C^*$-algebras, taking advantage of the former being Banach-Lie groups. For a given unital $C^*$-algebra $A$, we provide a description of the closed normal subgroup structure of the connected component of the identity of the unitary group, denoted by $U_A$, resp. of the subgroup of approximately inner automorphisms induced by the connected component of the identity of the unitary group, denoted by $V_A$, 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 $V_A$ 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 $V_A$ is perfect. We also characterize unital $C^*$-algebras $A$ such that $U_A$, resp. $V_A$ are topologically simple, generalizing the main results from [26]. 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 $C^*$-algebras of discrete groups and we show that when $A$ is a reduced group $C^*$-algebra of a non-amenable countable discrete group, then $A$ is simple if and only if $U_A/\mathbb{T}$ is topologically simple.