  We show that the commutator subgroup of the group of unitaries connected to
the identity in a simple unital C*-algebra is simple modulo its center. We then
go on to investigate the role of regularity properties in the structure of the
special unitary group of a C*-algebra. Under mild assumptions, we show that
this group has the invariant automatic continuity property and a unique polish
group topology. Strengthening our assumptions in the case of simple
C*-algebras, we show that the special unitary group modulo its center has
bounded normal generation. These results apply to all simple purely infinite
C*-algebras and too all simple nuclear C*-algebras in the "classifiable class".
We show with counterexamples how our conclusions may in general fail if no
regularity conditions are imposed on the C*-algebra.
