2
$\begingroup$

Let $G$ be a locally compact group. Put $I(G)=\ker: C^{*}(G) \to C_{r}^{*} (G)$, the kernel of the canonical morphism.

What type of $C^{*}$ algebras can not be isomorphic to $I(G)$, for some locally compact group $G$?In particular finit dimensional, unital, abelian,. etc? Is there any reference for characterization of such algebras?

Improve this question
$\endgroup$
5
  • 2
    $\begingroup$ This is a wide open question. I personally believe that, for $G$ discrete, the question is intractable, because a description of $I(G)$ is very much related to a description of the dual $\hat{G}$. As an example, consider a residually finite group with property (T). It is known that each finite-dimensional rep is isolated in $\hat{G}$, so defines a direct summand of $C^*(G)$ isomorphic to a matrix algebra. So $I(G)$ contains a huge direct sum of matrix algebras, and probably lots of other stuff too. $\endgroup$
    – Alain Valette
    Commented Jan 17, 2015 at 8:32
  • $\begingroup$ @AlainValette Prof. Valette, Thank you very much for your very interesting comment. $\endgroup$
    – Ali Taghavi
    Commented Jan 17, 2015 at 9:46
  • $\begingroup$ @AlainValette Can I ask you to give a reference for these materials? $\endgroup$
    – Ali Taghavi
    Commented Jan 17, 2015 at 18:55
  • $\begingroup$ @AlainValette I apologize for this second message. may you give a reference on material of your comment. thank you. $\endgroup$
    – Ali Taghavi
    Commented Jan 18, 2015 at 14:39
  • 1
    $\begingroup$ See "On Isolated Points in the Dual Spaces of Locally Compact Groups", by S. P. Wang, Mathematische Annalen (1975) Volume: 218, page 19-34; available on gdz.sub.uni-goettingen.de/dms/load/img/… $\endgroup$
    – Alain Valette
    Commented Jan 19, 2015 at 21:40

0

Reset to default

You must log in to answer this question.