2023/03/06 by Joan Claramunt, Claramunt, Joan, Łukasz Grabowski +1
Mathematics · #Advanced Operator Algebra Research #FOS: Mathematics #Group Theory (math.GR) #Holomorphic and Operator Theory #Operator Algebras (math.OA) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2303.02823
openalex publication_date 2023/03/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let Γ be a finitely generated torsion-free group. We show that the statement of Γ being virtually abelian is equivalent to the statement that the *-regular closure of the group ring ℂ[Γ] in the algebra of (unbounded) operators affiliated to the group von Neumann algebra is a central division algebra. More generally, for any field k, it is shown that k[Γ] embeds into a central division algebra in case Γ is virtually abelian. We take advantage of this result in order to develop a criterion for existence of units in the group ring k[Γ]. We develop this criterion in the particular case of Γ being the Promislow's group.