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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