On the algebraic tensor product space of $C^*$-algebra, I try to find an example whose maximal $C^*$-norm is not the minimal $C^*$-norm, but it seems as it is impossible to do this because the finite $C^*$-algebra is nuclear and so does the AF-algebra, so the $C^*$-norm has an interesting phenomenon, the maxiamal $C^*$-norm must be equal to minimal $C^*$-norm except one $C^*$-norm has no definition (the $C^*$-tensor product space is smaller).
Am I right? Does any name of interesting phenomenon and any this books or papers discuss this phenomenon?
if $A$ is a finite-dimensional and $B$ is any C$^*$-algebra, then the algebraic tensor $A\odot B$ is already a C$^*$-algebra, and so there is a single possible tensor norm.
On the other hand, singly generated is not enough to make a C$^*$-algebra AF, nor nuclear; not even exact (see the answer to this question). For any such $A$, as it is non-nuclear, there exists $B$ with $A\otimes_\min B\ne A\otimes_\max B$.