  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 $I_{n}$, $n < \infty$.
