Is there an example of simple and non-nuclear(non-amenable) $C^*$-algebra?
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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 PennigCommented Oct 20, 2015 at 8:19
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$\begingroup$ @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 ChoiCommented Oct 20, 2015 at 13:27
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1 Answer
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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).
answered Oct 20, 2015 at 15:33
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$\begingroup$ The only link I can find for Powers' paper is behind a paywall, sorry. $\endgroup$ Commented Oct 20, 2015 at 15:33
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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 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 :) $\endgroup$ Commented Oct 20, 2015 at 15:59
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1$\begingroup$ @YemonChoi: and yet you mentioned them ... $\endgroup$ Commented Oct 20, 2015 at 18:51
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1$\begingroup$ @NikWeaver Who says I have to mention pink elephants? Oh, damn :) $\endgroup$ Commented Oct 20, 2015 at 18:59
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1$\begingroup$ 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 $\endgroup$ Commented Oct 20, 2015 at 19:02