Group rings of infinite products of groups Given a infinite family of groups $(G_i)$ for $i\in I$. Is there a ring theoretic construction, that produces $R[\prod_{i\in I} G_i]$ using only the rings $(R[G_i])_{i\in I}$ ?
For the case of a finite family, we have $R[G\times H]\cong R[G][H]$ and for commutative $R$ we have $R[G\times H]\cong R[G]\otimes R[H]$. Neither of those constructions generalizes to the infinite case, e.g.
The map $R[\prod_i G_i]\rightarrow \mbox{invlim}_{I'\subset I, |I'|<\infty}R[\prod G_i]$ is not surjective (This product runs over $i\in I'$). The same holds for the map into the infinite tensor product (assuming that $R$ is commutative).
So I am hoping, that there is a better contruction in a more elaborate category (like $R$-Algebras with an augmentation), that produces $R[\prod_{i\in I} G_i]$ out of the group rings $(R[G_i])_{i\in I}$ .
 A: I'm guessing that $R[\prod_i G_i]$ might be obtained as the inverse limit you wrote in the category of Hopf algebras over $R$. Here the forgetful functor 
$$U: HopfAlg_R \to Coalg_R$$ 
preserves and reflects limits, so it suffices to check the claim in the category of (cocommutative) coalgebras. The guess then is that, by some application of the principle that every coalgebra is the filtered colimit of its finite-dimensional subcoalgebras, that the limit in $Coalg_R$ picks up only functions $\prod_i G_i \to R$ of finite support. 
See these slides for some hints on calculating limits of Hopf algebras.  
A: I started a work in 2009 concerning infinite tensor products of complex vector spaces, $^*$-algebras and Hilbert spaces. The resulting article has just been accepted for publication and I put it in the arxiv: http://arxiv.org/abs/1112.3128. In the article, it is shown in Example 3.1 that the infinite tensor product of $\mathbb{C}[G_i]$ is big enough to contain $\mathbb{C}[\prod_{i\in I} G_i]$. But in order to identify $\mathbb{C}[\prod_{i\in I} G_i]$, I still need to consider the group $\prod_{i\in I} G_i$. 
A: After some thoughts, I think your question has a positive answer if $R=\mathbb{C}$ (or $\mathbb{R}$) and one equips $\mathbb{C}[G_i]$ with the extra structure of the $\ell^1$-norm. By the way, your construction of an inverse limit also requires the extra structure of a fixed character on each $R[G_i]$ (in order to map $\bigotimes_{i\in I'} R[G_i]$ to $\bigotimes_{i\in I''} R[G_i]$, when $I''\subseteq I'$). 
