Normal subgroups of invertibles and of unitaries in a \(C^\ast\)-algebra (Q2332788)

This paper studies normal subgroups of the group of invertible and unitary elements in a \(C^\ast\)-algebra which lie in the connected component of the identity. The author develops a general description of closed normal subgroups in terms of closed two-sided ideals of the \(C^\ast\)-algebra. The group of unitary elements mentioned above leads to the group of approximately inner automorphisms of the \(C^\ast\)-algebra, and the author describes the closed normal subgroups of this group in a similar manner. As an application, the author answers a question of \textit{G. A. Elliot} and \textit{M. Rørdam} [Commun. Math. Phys. 155, No. 1, 3--26 (1993; Zbl 0895.46029)] by showing that in a simple \(C^\ast\)-algebra this group of approximately inner automorphisms is topologically simple. In addition, non-closed normal subgroups are discussed as well.