In principle, associative \*-algebras can be equipped with multiple norms. Can this be done in such a way that (after closure in the respective norm) they are turned into $C^\*$-algebras in multiple, not essentially equivalent ways?



Multiple $C^*$ structures?
There is at most one norm making an associative $\*$-algebra $\mathcal A$ into a C$^\*$-algebra, that is $\mathcal A$ is closed with respect to this C$^\*$-norm. This is because for all $a\in \mathcal A$, 
$$ \|a\|^2 = \|a^\*a\| = \textrm{spectral radius of}\: a^\*a = \textrm{sup}\{|\mu| : a^\*a - \mu I \notin \mathcal A^{-1}\},$$
that is, the norm of any element is determined by an algebraic property. The above is stated for unital $\mathcal A$ but it implies that the same holds true for non-unital as well.


However, if we only require that $\mathcal A$ is not norm closed with respect to this new norm, making $\mathcal A$ into a pre-C$^\*$-algebra then there are lots of potential C$^\*$-norms. For instance if $\mathcal A$ and $\mathcal B$ are C$^\*$-algebras then the algebraic tensor product $\mathcal A\odot \mathcal B$ is a $\*$-algebra with many potential norm structures. The most notable are $\mathcal A \otimes\_{min} \mathcal B$ and $\mathcal A \otimes\_{max} \mathcal B$, the min and max tensor products, of which there are examples where they are not $\*$-isomorphic.


