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



simple and non nuclear $C^*$-algebra
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 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](http://arxiv.org/abs/1410.2518)).


