Let $A$ and $B$ be two finite-dimensional C*-algebras. Let $\gamma$ denote the projective Banach space tensor product norm on the algebraic tensor product $A\odot B$, so $\gamma(t)=\inf\{\sum_{i}\|a_i\|\|b_i\|:t=\sum_i a_i\otimes b_i\}$. Then $\gamma(ts)\leq\gamma(t)\gamma(s)$ and $\gamma(t^*)=\gamma(t)$, but does $\gamma$ also satisfy $\gamma(t^*t)=\gamma(t)^2$?

I know that $\gamma$ should be equivalent to the C*-algebra norm on $A\odot B$, since all norms on finite-dimensional spaces are equivalent, but I would like to have equality, since this would imply that the embedding of finite-dimensional C*-algebras with subunital completely positive maps in the category of Banach spaces with contractions and with the projective tensor product as a monoidal product is a strong monoidal functor.