I'm currently reading the paper The ideal structure of the Haagerup tensor product of $C^{\ast}$-algebras and having difficulty in understanding the proof of Proposition $4.5$ from the paper.
Let $A$ and $B$ be $C^{\ast}$-algebras and $I$ be a nonzero closed ideal of $A \otimes^{\text{min}} B$, then $I$ contains a nonzero elementary tensor.
Suppose not. Let $\pi: A \otimes^{\text{min}} B \to \frac{A \otimes^{\text{min}} B}{I}$ be the natural map, and define a $C^{\ast}$-seminorm as $N(u) = \| \pi(u) \|$ for $ u \in A \otimes^{\text{min}} B$. By assumption, $N(a\otimes b) > 0$ for all nonzero elementary tensors $a \otimes b$.
Then it's written that $N$ norm dominates the min norm, but I don't understand how $N$ is a norm. I can only see that it's a seminorm.
P. S: Copy of the same question on mathoverflow can be found here.