2016/03/06 by Robert, Leonel
#FOS: Mathematics #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.1603.01884
We investigate the normal subgroups of the groups of invertibles and unitaries in the connected component of the identity. 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 the unitary case. We use this to prove a conjecture by Elliott and Rordam: in a simple 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 simple unital 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.