Is there an example of simple and nonnuclear(nonamenable) $C^*$algebra?

2$\begingroup$ Look for C*simple groups. For example, Powers has shown that the reduced group C*algebra of a nonabelian free group is simple. $\endgroup$ – Ulrich Pennig Oct 20 '15 at 8:19

$\begingroup$ @UlrichPennig Why not leave this as an answer? Although this is standard knowledge for specialists it strikes me that it would not be universally known $\endgroup$ – Yemon Choi Oct 20 '15 at 13:27
Following Yemon Choi's suggestion I turn my comment into an answer:
Lance gave a characterization of amenability in terms of the reduced group $C^*$algebra: A discrete group $G$ is amenable if and only if $C^*_r(G)$ is nuclear. Therefore one approach to finding examples of nonnuclear simple $C^*$algebras might be to look for nonamenable groups, such that their reduced group $C^*$algebra is simple.
Powers showed in the paper "Simplicity of the $C^*$algebra associated with the free group on two generators" that this is in fact true for $G = \mathbb{F}_2$. But the story does not end here. A lot of interesting research has been done concerning the question of $C^*$simplicity of discrete groups. You might want to look for example at the fairly recent paper of Breuillard, Kalantar, Kennedy and Ozawa (arXiv).

$\begingroup$ The only link I can find for Powers' paper is behind a paywall, sorry. $\endgroup$ – Ulrich Pennig Oct 20 '15 at 15:33

3$\begingroup$ Just because it is something I always go on about, I shall do so here :) ... Lance proved that $G$ is amenable iff ${\rm C}_r^*(G)$ is nuclear, using one of the original definitions of nuclearity. Then Bunce proved, without requiring any of Lance's results, that if ${\rm C}_r^*(G)$ is amenable as a Banach algebra and $G$ is discrete, then $\ell^\infty(G)$ has a (left or right or twosided) invariant mean, so $G$ is amenable. I mention these in order to emphasize that nonnuclearity & nonamenability of ${\rm C}_r^*(F_2)$ can be discussed without mentioning the names of Connes or Haagerup :) $\endgroup$ – Yemon Choi Oct 20 '15 at 15:59

$\begingroup$ @YemonChoi: and yet you mentioned them ... $\endgroup$ – Nik Weaver Oct 20 '15 at 18:51

$\begingroup$ @NikWeaver Who says I have to mention pink elephants? Oh, damn :) $\endgroup$ – Yemon Choi Oct 20 '15 at 18:59

$\begingroup$ BTW I hope my comment wasn't perceived as denigrating the amenable iff nuclear results, which I hold in very high regard. It's just that sometimes they are used when there are easier, and more informative, approaches for particular examples one is considering $\endgroup$ – Yemon Choi Oct 20 '15 at 19:02