Cross norms on tensor product of universal $C^*$-algebras and Kirchberg property 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
 A: 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.
A: 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.
A: Wiersma solved that problem finding many groups with inequality. 
