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

TL;DR

This paper investigates the structure and topological properties of the unitary groups in simple unital C*-algebras, establishing simplicity, automatic continuity, and bounded normal generation under certain regularity conditions.

Contribution

It demonstrates the simplicity of the commutator subgroup modulo its center and proves automatic continuity and bounded normal generation for the special unitary group in specific C*-algebras.

Findings

Commutator subgroup of unitaries connected to identity is simple modulo center.

Special unitary group has automatic continuity under mild regularity assumptions.

Bounded normal generation holds for simple C*-algebras with strengthened assumptions.

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 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.