Let $G$ be a locally compact group. Is there any correspondence between closed two-sided ideals of $L^1(G)$ and closed normal subgroups of $G$? (Especially, is there any correspondence between finite codimension closed two-sided ideals of $L^1(G)$ and closed normal subgroups of $G$?)



Ideals of $L^1(G)$ and normal subgroups of $G$
The following is about the best you can hope for, I think.


For discrete groups a homomorphism between groups gives rise to a homomorphism between algebras. So although we do not have a "correspondence", in this setting a normal subgroup $N\lhd G$ gives rise to an ideal $J\_N:= \ker( \ell^1(G) \to \ell^1(G/N))$. A simple cardinality argument shows you that $J\_N$ is usually much bigger than $\ell^1(N)$.


As I said in my comment, in general there can be many more ideals.


For locally compact groups, the functorial properties of $G\mapsto L^1(G)$ are not so good. But if I recall correctly, when $N\lhd G$ is a *closed* normal subgroup then we still obtain a well-defined, continuous algebra homomorphism $L^1(G)\to L^1(G/N)$, and hence its kernel $J\_N$ will still be a closed $2$-sided ideal in $L^1(G)$.


