If $\mathfrak S$ denotes the set of all non-zero C$^*$-algebras up to $*$-isomorphism, then $(\mathfrak S, \otimes_\textrm{min})$ and $(\mathfrak S, \otimes_\textrm{max})$ are abelian semigroups with identity element $\mathbb C$ (or commutative monoids if you prefer that language) since both tensor products are commutative and associative.

> **Question:** Is there any cancellation in these semigroups? Does there exist $\mathcal C\in \mathfrak S$ not isomorphic to $\mathbb C$ such that for all $A,B\in \mathfrak S$ we have
$$
\mathcal A \otimes_\textrm{min} \mathcal C \simeq \mathcal B\otimes_\textrm{min} \mathcal C \Rightarrow \mathcal A \simeq \mathcal B
$$
or the similar statement with $\otimes_\textrm{max}$?