# A non nuclear $C^*$ algebra $A$ for which the algebraic tensor product $A\otimes A$ admits a unique $C^*$ norm

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 $$A\otimes_{\max{}} A = A\otimes_{\min{}} A$$.