Is there a non nuclear $C^*$ algebra $A$ for which the minimum and maximum $C^*$ norms on $A\otimes A$ coincide?
A somewhat similar question is discussed here.
Is there a non nuclear $C^*$ algebra $A$ for which the minimum and maximum $C^*$ norms on $A\otimes A$ coincide?
A somewhat similar question is discussed here.
Pisier https://arxiv.org/abs/1908.02705 very recently constructed a non-nuclear $C^\ast$-algebra $A$ with the weak expectation property (WEP) and the local lifting property (LLP). By a celebrated result of Kirchberg (see Corollary 13.2.5 in Brown and Ozawa's book) it follows that if $B$ has WEP and $C$ has LLP, then $B\otimes_{\max{}} C = B\otimes_{\min{}} C$. Hence $A\otimes_{\max{}} A = A\otimes_{\min{}} A$.