Normal subgroups of invertibles and of unitaries in a C*-algebra

TL;DR

This paper characterizes the structure of normal subgroups in the invertible and unitary groups of a C*-algebra, proving a conjecture and revealing simplicity properties of certain automorphism groups.

Contribution

It establishes a 'sandwich condition' linking normal subgroups to ideals and proves the topological simplicity of approximately inner automorphisms in simple C*-algebras.

Findings

Normal subgroups correspond to closed two-sided ideals.

Proved the Elliott-Rordam conjecture for simple C*-algebras.

The commutator subgroup of invertibles is simple modulo its center.

Abstract

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.