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Does anyone know, if the following result has been proved ?

Let G be an infinite discrete group. A = L1(G) it's algebra and Prim(A) the set of prime ideals with spectral topology. The result is : faPrim(A) is Hausdorff if and only if G is a FC-Group.

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    $\begingroup$ I should know this, but: is your condition equivalent to G being a Type I group? (It feels stronger but I can't immediately think of a counter-example) $\endgroup$
    – Yemon Choi
    Commented Aug 12, 2012 at 20:11
  • $\begingroup$ If true, this feels like something that Kaniuth might have worked on... I will try to look up details when I next have a free moment in the office. $\endgroup$
    – Yemon Choi
    Commented Aug 12, 2012 at 21:19
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    $\begingroup$ @Yemon: the infinite dihedral group is type I, but $Prim(A)$ is not Hausdorff ... $\endgroup$
    – Alain Valette
    Commented Aug 13, 2012 at 15:04
  • $\begingroup$ Merci Alain - I should have remembered that! $\endgroup$
    – Yemon Choi
    Commented Aug 13, 2012 at 19:12
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    $\begingroup$ Just to clarify: I've usually seen Prim(A) used to denote the primitive ideal space. Do you really mean "prime ideals"? $\endgroup$
    – Yemon Choi
    Commented Aug 13, 2012 at 23:51

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I asked Eberhard Kaniuth, if he knows something about the problem. Here is his answer :

If G is a FC-Group, then Prim(L^1(G)) is Hausdorff. If Prim(L^1(G)) is Hausdorff and G amenable, then G is a FC-Group Probably G need not to be amenable. He don't know any counterexample.

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