Maximal Discrete Subgroups in Unitary Groups of Operator Algebras

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 utilize free probability theory. In addition, we clarify the situation for 𝐶*-algebras.