Geometric properties of unit groups of von Neumann's continuous rings
Abstract
We prove that, if $R$ is a non-discrete irreducible, continuous ring, then its unit group $\mathrm{GL}(R)$, equipped with the topology generated by the rank metric, is topologically simple modulo its center, path-connected, locally path-connected, bounded in the sense of Bourbaki, and not admitting any non-zero escape function. All these topological insights are consequences of more refined geometric results concerning the rank metric, in particular with regard to the set of algebraic elements. Thanks to the phenomenon of automatic continuity, our results also have non-trivial ramifications for the underlying abstract groups.
Fields of Study & Subjects
Advanced Topology and Set Theory
Advanced Operator Algebra Research
Rings
Modules
and Algebras
Publication details
- Journal / source
- Journal of Algebra
- Volume
- 712
- Pages
- 654–694
- Publication date
- 1 September 2026
- Language
- EN