Group von Neumann algebras, inner amenability, and unit groups of continuous rings
Friedrich Martin Schneider
TL;DR
The paper investigates how inner amenability of a discrete group $G$ governs the amenability of the unit group of the ring of affiliated operators $\mathrm{R}(\mathrm{N}(G))$ in the rank topology. By developing an Eymard–Greenleaf amenability framework for actions on spaces of projections and exploiting a Lipschitz, equivariant correspondence between projection lattices and affiliated operator rings, the author links group-theoretic properties to topological dynamics of unit groups. The main result shows that if $G$ is not inner amenable, then $\mathrm{GL}(\mathrm{R}(\mathrm{N}(G)))_{\mathrm{rk}}$ is non-amenable, producing non-discrete irreducible, continuous rings with non-amenable unit groups and connecting this to actions of $G$ on projection spaces. This leads to concrete examples (e.g., free groups and Thompson groups) and establishes a criterion connecting inner amenability to amenability of rank-topology unit groups, enriching the interaction between operator algebras and group-theoretic properties.
Abstract
We prove that, if a discrete group $G$ is not inner amenable, then the unit group of the ring of operators affiliated with the group von Neumann algebra of $G$ 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 $\mathrm{II}_{1}$ factor on the associated space of projections of a fixed trace.