A C*-algebra $A$ is called nuclear iff there is a unique cross C*-norm on the algebraic tensor product $A\otimes B$ for any other C*-algebra $B$. It is equivalent that the minimal (spatial) and maximal norms on $A\otimes B$ coincide.
I know (cf. this MathOverflow question) that an example of a non-nuclear C*-algebra ist $C_r^\ast(\mathbb F_2)$ where $\mathbb F_2$ is the free group on two generators and $C_r^\ast(\mathbb F_2)$ denotes its reduced group C*-algebra.
Now my question is: Is there an explicit example of a C*-algebra $B$ such that the minimal and maximal norms on $C_r^\ast(\mathbb F_2)\otimes B$ do not coincide? With explicit I mean that one can write down both norms and see that they are different, e.g. by evaluating them on some specific element. Are there easier examples for other C*-algebras instead of $C_r^\ast(\mathbb F_2)$?