Normal subgroups of invertibles and of unitaries in a C*-algebra
AbstractWe investigate the normal subgroups of the groups of invertibles and unitaries in the connected component of the identity of a {\mathrm{C}^{*}}-algebra. By relating normal subgroups to closed two-sided ideals we obtain a “sandwich condition” describing all the closed normal subgroups both in the invertible and in the unitary case. We use this to prove a conjecture by Elliott and Rørdam: in a simple \mathrm{C}^{*}-algebra, the group of approximately inner automorphisms induced by unitaries in the connected component of the identity is topologically simple. Turning to non-closed subgroups, we show, among other things, that in a simple unital \mathrm{C}^{*}-algebra the commutator subgroup of the group of invertibles in the connected component of the identity is a simple group modulo its center. A similar result holds for unitaries under a mild extra assumption.