Let $A$ and $B$ be two C*-algebras. Then $(A,B)$ is called is a nuclear pair if there is a unique $C^*$-norm on the algebraic tensor product $A\odot B$.
If $A$ or $B$ is nuclear, then all pairs $(A,B)$ are nuclear. But there are many examples of non-nuclear C*-algebras building nuclear pairs. One classical example is $A=B(\ell^2)$ and $B=C^*(\mathbb{F})$, for $\mathbb{F}$ a noncommutative free group and $\ell^2:=\ell^2(\mathbb N)$.
Usually, when $(A,B)$ is not a nuclear pair, then one can find (infinitely) many C*-norms on $A\odot B$. This is the case for $B(\ell^2)\odot B(\ell^2)$.
My question is the following: suppose $(A,B)$ is not is a nuclear pair, that is, the maximal and the minimal $C^*$-norms on $A\odot B$ are different. Is there always then another $C^*$-norm on $A\odot B$ which is different from both the maximal and the minimal $C^*$-norms?
