On the geometry of idempotents in von Neumann algebras

TL;DR

This paper explores the structure of idempotents in von Neumann algebras through the lens of the general linear group, providing characterizations and bijections that extend previous work on regular rings.

Contribution

It characterizes sets of idempotents in von Neumann factors using properties of the general linear group and establishes bijections preserving right ideal generation.

Findings

Characterization of idempotents via the general linear group

Bijection of general linear groups induces idempotent bijections

Extension of regular ring results to von Neumann factors

Abstract

We consider the general linear group as an invariant of von Neumann factors. We prove that up to complement, a set consisting of all idempotents generating the same right ideal admits a characterisation in terms of properties of the general linear group of a von Neumann factor. We prove that for two Neumann factors, any bijection of their general linear groups induces a bijection of their idempotents with the following additional property: If two idempotents or their two complements generate the same right ideal, then so does their image. This generalises work on regular rings, such include von Neumann factors of type $I_{n}$, $n < \infty$.