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arXiv · 2108.09796

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

Abhinav Chand·Leonel Robert
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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.

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Abhinav Chand, Leonel Robert. 2021-08-22. Simplicity, bounded normal generation, and automatic continuity of groups of unitaries. https://arxiv.org/abs/2108.09796

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