Simplicity, bounded normal generation, and automatic continuity of groups of unitaries

Abstract

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 to 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.