# simple and non nuclear $C^*$-algebra

Is there an example of simple and non-nuclear(non-amenable) $C^*$-algebra?

• Look for C*-simple groups. For example, Powers has shown that the reduced group C*-algebra of a nonabelian free group is simple. – Ulrich Pennig Oct 20 '15 at 8:19
• @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 – Yemon Choi Oct 20 '15 at 13:27

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 non-nuclear simple $C^*$-algebras might be to look for non-amenable 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).
• 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 two-sided) invariant mean, so $G$ is amenable. I mention these in order to emphasize that non-nuclearity & non-amenability of ${\rm C}_r^*(F_2)$ can be discussed without mentioning the names of Connes or Haagerup :) – Yemon Choi Oct 20 '15 at 15:59