I want to study the closed ideal structure of $L^1(G)$. Is there a good paper or book which characterizes closed ideals and maximal ideals of $L^1(G)$?
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$\begingroup$ If you complete $L^1(G)$ to get the C*-algebra $A$, the maximal ideal spectrum of $A$ is the unitary dual of $G$. $\endgroup$– user1688Commented May 31, 2016 at 7:08
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1$\begingroup$ For abelian groups there was a lot of work in the 1960s and 1970s: some discussion is in Rudin's book, I think. But you really should specify what you want to know about closed ideals. One should really think of questions before asking about the literature $\endgroup$– Yemon ChoiCommented May 31, 2016 at 13:43
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1$\begingroup$ I mean 2-sided maximal ideal. I want to know is there any classification about 2-sided closed ideals in $L^1(G)$? Or any identification aboat closed 2-sided ideals with the finite codimention?($G$ is not an abelian group) $\endgroup$– Albert haroldCommented May 31, 2016 at 15:26
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2$\begingroup$ Mr/Mrs pseudonym: the study of closed 2-sided ideals in the L^1 group algebra is interesting but I think in general very diificult. Note that even in the abelian case we don't have a complete classification, because (viewing $L^1(G)$ as $A(\hat{G})$, for G abelian) there are different ideals in $A(\hat{G})$ that have the same hull (I think hulls and kernels must be discussed in the recent Banach algebras book of Kaniuth). This is connected to the problem of determining which closed sets are sets of synthesis. In the non-abelian world things could be even wilder. $\endgroup$– Yemon ChoiCommented May 31, 2016 at 19:55
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4$\begingroup$ As a general point, a very useful source if you want to see what is known about group algebras, is Volume 2 of Palmer's book(s) on Banach algebras. It does not always have full proofs or give the most complete results, but it gives many references that one can try to look up, which may suggest ideas for research $\endgroup$– Yemon ChoiCommented May 31, 2016 at 19:59
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For abelian case, close ideal of $L^1(G)$ is equal to translation invariant subspace of $L^1(G)$.[ please see: W.Rudin, Harmonic analysis on semigroup] In general you can see [ Folland, Harmonic analysis]
