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Is there a way to characterize which complex algebras arise as the group algebra of some locally compact group? To make this more concrete, say $A$ is a sub-algebra of $\text{Mat}(n,\mathbb{C})$, $n\in\mathbb{N}$. Are there any conditions under which there exist criterion to determine whether $A$ is the group algebra of some subgroup of $GL(n,\mathbb{C})$? I am also interested in similar results for finite groups.

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When you say group algebra, are you completing this in some norm? (Otherwise your mention of "locally compact groups" sounds strange.) If so, do you mean $L^1(G)$, or $C_r^*(G)$, or $C^*(G)$, or something else? – Yemon Choi Sep 22 '11 at 20:42
I mean $C_c(G)$ - continuous compactly supported functions. Thanks for the clarification. – lwassink Sep 22 '11 at 20:48
The paper… of Rieffel characterizes group algebras of groups over the reals as ordered algebras (ordering functions on the group pointwise). – Benjamin Steinberg Sep 22 '11 at 22:01
That is finite groups. – Benjamin Steinberg Sep 22 '11 at 22:01
If the group is locally compact but not finite, isn't it impossible for $C_c(G)$ to be finite-dimensional? – MTS Sep 22 '11 at 23:58

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