On ergodicity of linear actions on $\mathbb{R}^n$ and factoriality of group von Neumann algebras
Soham Chakraborty, Chinmay Tamhankar
TL;DR
The article analyzes when semidirect products G = N ⋊ Γ have factorial group von Neumann algebras L(G), by translating factoriality into ergodicity properties of the dual action on N. Focusing on abelian Lie-type N, it reduces the problem to ergodicity and (dual) double ergodicity/mixing of the dual action Γ^T on $\mathbb{R}^n$, yielding that L(G) is a factor under natural diagonal-splitting conditions (and conversely under broad necessary conditions). It furnishes large classes of examples, notably with Γ a lattice or dense subgroup of SL(n, R) acting on R^n, and with discrete D having infinite nontrivial orbits, where the resulting semidirect product von Neumann algebras are factors (often II$_1$ or II$_ fty$). The results extend to torsion D where the cocycle vanishes, and they connect to broader factoriality criteria via crossing with dual actions and cocycle rigidity methods. Altogether, the work provides a concrete framework to produce factorial locally compact group von Neumann algebras from linear and diagonal actions on abelian Lie-type groups, enriching the catalog of explicit factor examples and clarifying the ergodic mechanisms behind factoriality.
Abstract
We give some natural conditions on actions of discrete countable groups on abelian locally compact groups of Lie type that imply factoriality of the group von Neumann algebras of their semidirect products. This allows us to give a fairly large class of examples of locally compact groups whose group von Neumann algebras are factors.