Suppose $A$ and $B$ are two $C^*$-algebras. On the algebraic tensor product $A\otimes B$ we can define the maximal and minimal tensor norms which makes $A\otimes B$ a $C^*$- algebra.
what are other possible norms which one define on algebraic tensor product of $C^*-$ algebras to make it again a $C^*$ -algebra?
The books I have seen only discusses these two norms while they do discuss many norms on tensor product of operator spaces.Why so?
Because, as far as I can tell, there is no general recipe to produce a C$^*$-norm on $A\otimes B$ other than the max and the min norms. And for all nuclear algebras $A$ you have that the max and the min norm are equal, so in general you cannot expect to find a "different" one.
With operator spaces and operator systems, as opposed to the case with C$^*$-algebras, there seems to be several natural ways to produce tensor norms.