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Let $G$ be a locally compact group and let $C^* (G)$ be the universal $C^*$-algebra of $G$; i.e., the completion of the convolution algebra $L^1(G)$ with respect to the norm $||f||=\sup_\pi||\pi(f)||$, where $\pi$ runs over all non-degenerate representations of $L^1(G)$ (seen as an involutive Banach algebra) in some $B(H)$.

Question: Is it known any example of a countable discrete non-amenable group such that $$ C^* (G)\otimes_{\min}C^* (G)=C^* (G)\otimes_{\max}C^* (G) $$

Edit: the original question was for general locally compact groups but, as observed by Scott and Yemon, $SL(2,\mathbb C)$ already works as an example, as well as any connected group. In relation to Connes' problem (see below), I am more interested in countable (and discrete) groups.

For $G=\mathbb F_\infty$, the statement above is equivalent to Connes' embedding conjecture, by an unexpected and beautiful theorem of Eberhard Kirchberg (Inventiones Math. 1994). Somebody asked me the previous question during a talk and, to be honest, I have no idea.

After talking with some people, it seems that this problem is open in both directions

Question 2 Is it known any example of a countable discrete group such that $$ C^* (G)\otimes_{\min}C^* (G)\neq C^* (G)\otimes_{\max}C^* (G) $$

Thanks in advance,

Valerio

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@Valerio: in order to avoid clogging up the comments to Scott L's answer, it may be best if you email me some time next week (regarding VN(G) being injective when G is connected) and I will try to dig through my collection of PDFs. I have Palmer volume 2 open in front of me and I cannot find the reference which pm has suggested in the comments below. –  Yemon Choi Mar 9 '12 at 23:04
    
Thank you, Yemon. I'll do it tomorrow. –  Valerio Capraro Mar 10 '12 at 3:13

2 Answers 2

In Lance's 1972 paper "On Nuclear $C^\ast$-algebras," it is mentioned that there are non-amenable groups for which $C^\ast(G)$ is nuclear. The main goal of the paper is to prove that when $G$ is discrete, then $C^\ast(G)$ is nuclear if and only if $G$ is amenable, but he does mention very briefly that in the general case, $C^\ast(G)$ can be nuclear even for a non-amenable group. In particular, he mentions $\rm{SL}(2, \mathbb{C})$ as a specific example.

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Perfect, I think that $SL(2,\mathbb C)$ answers the original question. The question looks apparently harder for countable discrete groups. Indeed, Kirchberg's property is quite weaker than nuclearity. –  Valerio Capraro Mar 9 '12 at 14:48
2  
You could take G to be any connected group - this follows from Connes's observation that VN of such a group is injective, although when G is Lie the result is really implicitly due to Dixmier/Pukanszky, and when G is semi simple Lie, its $C^\ast$-algebra is Type I by results of Harish-Chandra/Godement. The fact that SL(2,C) is Type I probably goes back to Bargmann or Gelfand, I forget. If self-advertising is not too gauche, see the remarks here ifwisdomwereteachable.wordpress.com/2012/02/03/… –  Yemon Choi Mar 9 '12 at 16:25
    
Thank you very much, Yemon, for this interesting comment. So, the question is really interesting for countable groups.. –  Valerio Capraro Mar 9 '12 at 16:36
    
By the way, I would be interested to see the proof that the von Neumann algebra of a connected group is injective. Could you please give a reference? –  Valerio Capraro Mar 9 '12 at 16:43
    
There is a slight misstatement in what I wrote - VN of a connected group is injective, but I'm not sure this will make the full C-star algebra nuclear. However this last claim is true for G semisimple Lie, as I said above. –  Yemon Choi Mar 9 '12 at 16:55

I do not know any example (with discrete $G$), but I can at least say that if such example is known to exist, it is nontrivial. Indeed, it was a long-standing open problem to find a nonnuclear $C^*$ algebra $A$ such that $A\otimes A^{op}$ carries only one $C^*$-norm. It is (one of) the main result(s) of the paper by Kirchberg that you cite that such a $C^*$-algebra exists (but the example is not a group $C^*$-algebra). My reference for this is Pisier's Introduction to operator space theory, Chapter 22.

Since $C^*(G)$ is always isomorphic to $C^*(G)^{op}$, you are asking whether $A$ can be taken as the full $C^*$-algebra of a discrete group. This refined question is not answered by Kirchberg, and a rapid bibliographic search did not give anything.

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Thank you very much, Mikael, for this interesting comment. –  Valerio Capraro Mar 10 '12 at 18:23

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