TITLE: Reference for Kaplansky's proof that in $\mathbb{C}[G]$, $ab=1$ implies $ba=1$
QUESTION [9 upvotes]: Here on the Wikipedia page for Group Rings, talking about group rings over infinite groups,

The case [of a group ring $R[G]$ where $G$ is an infinite group] where $R$ is the field of complex numbers is probably the one best studied. In this case, Irving Kaplansky proved that if $a$ and $b$ are elements of $\mathbf{C}[G]$ with $ab = 1$, then $ba = 1$. Whether this is true if $R$ is a field of positive characteristic remains unknown.

Wikipedia doesn't list a reference for this specifically though. Does anyone have a reference for this proof? I would appreciate any current related information about this result too.

REPLY [2 votes]: Perhaps the following answer your question, or will at least be of use.
Berlai, GROUPS SATISFYING KAPLANSKY’S STABLE FINITENESS CONJECTURE, p.1 provides:

Then Lupini, Lecture 201, p.4-5 provides: