Let $\Gamma$ be a group, $N(\Gamma)$ its group von Neumann algebra, $\newcommand{\cUG}{{\mathcal U}(\Gamma)}$ and $\cUG$ the ring of all densely-defined, closed operators $\ell^2(\Gamma)\to\ell^2(\Gamma)$ that are affiliated to $N(\Gamma)$.

In some work I've been doing on (algebras of) convolution operators associated to group representations, I'd like to give a clear reference for the fact that $\cUG$ is *directly finite*, or in alternative terminology *Dedekind finite*: this just means that whenever $a,b$ belong to $\cUG$ and satisfy $ab=1$, then $ba=1$.
(Note: I am only interested in the case of $\cUG$ but the analogous result holds if $N(\Gamma)$ is replaced by any other finite von Neumann algebra.)

The statement of this result is something I learned years ago from reading preprints of Elek, but there he gives as his references an article of Berberian, which in turn relies on various results about Baer $\ast$-rings in Berberian's own book on the same ... so this begins to look less and less like a convenient reference for readers who aren't already happy with von Neumann regular rings, Baer $\ast$-rings, and so on.

In contrast, I've found by doing some foraging online that the fact I need to cite follows straightforwardly from the polar decomposition for unbounded affiliated operators, as explained simply in Lemma 2.2 of this paper by H. Reich:

On the K- and L-theory of the algebra of operators affiliated to a finite von Neumann algebra. K-Theory, 24 (2001), no.4, 303-326 (Copy on author's webpage)

It seems odd to give the preliminary section of a 2001 paper as the citation/reference for something that must surely have been known to Murray and von Neumann. So my question is this:

*Can anyone suggest an older reference*, accessible for analysts and preferably in a book, for the fact that $ab=1 \implies ba=1$ for all $a,b\in\cUG$? Or is a quick outline, and a citation of Reich's exposition, the best that I can do?
I haven't had time to check in Kadison-Ringrose, nor Lück's book on $L^2$-invariants, but I can think of several MO regulars who would be much more familiar with the literature on $\cUG$ than I am.