2023/12/04 by Hiroshi Ando, Michal Doucha, Ando, Hiroshi +1
Mathematics · #Advanced Banach Space Theory #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Group Theory (math.GR) #Operator Algebras (math.OA)
paper · pdf · doi:10.48550/arxiv.2312.01794
openalex publication_date 2023/12/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 UA, resp. of the subgroup of approximately inner automorphisms induced by the connected component of the identity of the unitary group, denoted by VA, 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 VA 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 VA is perfect. We also characterize unital C^*-algebras A such that UA, resp. VA 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 UA/\mathbbT is topologically simple.