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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: This question was first posted on MSE.

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  • $\begingroup$ If one of the two $C^*$-algebras is exact, this is true. See corollary 9.4.6 in the book "C*-algebras and finite-dimensional approximations" by Brown-Ozawa. $\endgroup$
    – J. De Ro
    Commented May 23, 2023 at 19:42
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    $\begingroup$ An important part of their argument missing from your post is the reference to Takesaki's paper. Did you miss the reference or do you not understand how the claim follows form Takesaki's paper? $\endgroup$
    – Caleb Eckhardt
    Commented May 23, 2023 at 21:42
  • $\begingroup$ Dear @CalebEckhardt: I don't understand how it follows from Takesaki's paper. Can you please elaborate? Regards, $\endgroup$
    – Math Lover
    Commented May 23, 2023 at 22:46
  • $\begingroup$ Does the argument that N dominates the min-norm use that N is actually a norm? Because otherwise you can conclude from the fact that the min norm is a norm. $\endgroup$
    – J. De Ro
    Commented Jul 14 at 18:27

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