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.
