# Norms on Tensor Product of $C^*$- algebras

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.