Group von Neumann algebras, inner amenability, and unit groups of continuous rings
arXiv:2211.03537 · doi:10.1093/imrn/rnad181
Abstract
We prove that, if a discrete group is not inner amenable, then the unit group of the ring of operators affiliated with the group von Neumann algebra of is non-amenable with respect to the topology generated by its rank metric. This provides examples of non-discrete irreducible, continuous rings (in von Neumann's sense) whose unit groups are non-amenable with regard to the rank topology. Our argument establishes and uses connections with Eymard--Greenleaf amenability of the action of the unitary group of a factor on the associated space of projections of a fixed trace.
15 pages, no figures; v2: referee report taken into account, 17 pages