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 acountable discrete non-amenablegroup 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 2Is it known any example of acountable discretegroup such that $$ C^* (G)\otimes_{\min}C^* (G)\neq C^* (G)\otimes_{\max}C^* (G) $$

Thanks in advance,

Valerio