Uniqueness of C* norm and tensor product of C* algebra

Question

I am little bit confuse about the norms defined on tensor product of C* algebra. We can define spatial norm and maximal norm on $A\otimes B$ for $A,B$ are C* algebras. However there is theorem telling us that C* norm is unique on an algebra, what am I missing here?

Answer 1

You are misinterpreting the theorem. The theorem says:

Theorem: There is at most one $C^*$-norm on a $*$-algebra $A$ such that $A$ becomes a $C^*$-algebra.

In general, a $*$-algebra can carry a lot of $C^*$-norms. The subtlety here is that a $*$-algebra together with a $C^*$-norm is not a $C^*$-algebra: the $*$-algebra need not be a complete normed space with respect to this $*$-norm.

Given two $C^*$-algebra $A$ and $B$, we can form the algebraic tensor product $A \odot B$. If $A$ and $B$ are both infinite-dimensional, $A \odot B$ will in general not be complete and thus not a $C^*$-algebra.