Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 18 (2022), 052, 7 pages      arXiv:2206.06704      https://doi.org/10.3842/SIGMA.2022.052
Contribution to the Special Issue on Non-Commutative Algebra, Probability and Analysis in Action

Maximal Discrete Subgroups in Unitary Groups of Operator Algebras

Vadim Alekseev and Andreas Thom
TU Dresden, Institut für Geometrie, 01062 Dresden, Germany

Received February 22, 2022, in final form June 29, 2022; Published online July 09, 2022

Abstract
We show that if a group $G$ is mixed-identity-free, then the projective unitary group of its group von Neumann algebra contains a maximal discrete subgroup containing $G$. The proofs are elementary and make use of free probability theory. In addition, we clarify the situation for $C^*$-algebras.

Key words: maximal discrete subgroups; unitary groups; operator algebras.

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