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Given a group $G$ we may consider its group ring $\mathbb C[G]$ consisting of all finitely supported functions $f\colon G\to\mathbb C$ with pointwise addition and convolution. Take $f,g\in\mathbb C[G]$ such that $f*g=1$. Does this imply that $g*f=1$?

If $G$ is abelian, its group ring is commutative, so the assertion holds. In the non-abelian case we have $f*g(x)=\sum_y f(xy^{-1})g(y)$, while $g*f(x)=\sum_y f(y^{-1}x)g(y)$, and this doesn't seem very helpful.

If $G$ is finite, $\dim_{\mathbb C} \mathbb C[G]= |G|<\infty$, and we may consider a linear operator $T\colon \mathbb C[G]\to\mathbb C[G]$ defined by $T(h) = f*h$. It is obviously surjective, and hence also injective. Now, the assertion follows from $T(g*f)=f=T(1)$.

What about infinite non-abelian groups? Is a general proof or a counterexample known?

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up vote 20 down vote accepted

A ring is called Dedekind-finite if that property holds. Semisimple rings are Dedekind finite, so this covers $\mathbb CG$ for a finite group $G$; this is easy to do by hand. It is a theorem of Kaplansky that this also holds $KG$ for arbitrary groups $G$ and arbitrary fields $K$ of characteristic zero. See [Kaplansky, Irving. Fields and rings. The University of Chicago Press, Chicago, Ill.-London 1969 ix+198 pp. MR0269449] It is open, I think, for general fields.

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"...if $R$ is an Artinian ring and $G$ is a sofic group, then the group ring $R[G]$ is stably finite..." -MR2362939 (2009a:16046) Ceccherini-Silberstein, Tullio(I-SAN-EN); Coornaert, Michel(F-STRAS-I) Linear cellular automata over modules of finite length and stable finiteness of group rings. J. Algebra 317 (2007), no. 2, 743--758. See also MR2335561 (2008j:16077) Nasrutdinov, M. F.(RS-KAZA) Stable finiteness of group rings. (Russian) Izv. Vyssh. Uchebn. Zaved. Mat. 2006, , no. 11, 29--32; translation in Russian Math. (Iz. VUZ) 50 (2006), no. 11, 27--30 (2007) –  Jonas Meyer Mar 17 '10 at 18:42
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It is worth noting that (AFAIK) there is no group which is known to be non-sofic. Also, in more or less the same breath, Kaplansky observes that the group von Neumann algebra of a discrete group, like any finite von Neumann algebra, is Dedekind finite (also sometimes called directly finite). A slightly lower tech proof can be extracted from M. S. Montgomery, Left and right inverses in group algebras, Bull. Amer. Math. Soc. 75 (1969) 539--540. (Might also shamelessly plug arxiv.org/abs/1003.1650 while I'm typing...) –  Yemon Choi Mar 17 '10 at 20:16
    
One should add that the result for sofic groups was first shown by Elek and Szabo in "Sofic groups and direct finiteness", Journal of Algebra Vol 280, Issue 2 –  Łukasz Grabowski Jun 15 '11 at 11:50

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