Abhinav Chand | Searcharxiv

arXiv2021

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

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.

math.OA↗

arXiv2020

Simultaneous averaging to zero by unitary mixing operators

We show that if every element a vector subspace of a C*-algebra can be averaged to zero by means of unitary mixing operators, then all the elements of the subspace can be simultaneously averaged to zero by a net of unitary mixing operators. Moreover, such subspaces admit a simple description in terms of commutators and kernels of states on the C*-algebra. We apply this result to center-valued expectations in C*-algebras with the Dixmier property.

math.OA↗