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 subalgebra 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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$\begingroup$ 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? $\endgroup$– Yemon ChoiCommented Sep 22, 2011 at 20:42

$\begingroup$ I mean $C_c(G)$  continuous compactly supported functions. Thanks for the clarification. $\endgroup$– lwassinkCommented Sep 22, 2011 at 20:48

2$\begingroup$ The paper projecteuclid.org/… of Rieffel characterizes group algebras of groups over the reals as ordered algebras (ordering functions on the group pointwise). $\endgroup$– Benjamin SteinbergCommented Sep 22, 2011 at 22:01

$\begingroup$ That is finite groups. $\endgroup$– Benjamin SteinbergCommented Sep 22, 2011 at 22:01

3$\begingroup$ If the group is locally compact but not finite, isn't it impossible for $C_c(G)$ to be finitedimensional? $\endgroup$– MTSCommented Sep 22, 2011 at 23:58
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