Let $A$ be an algebra over $\mathbf{C}$ with an involution operator, and let $\|\cdot\|_1$ and $\|\cdot\|_2$ be two $equivalent$ operator norms, making $A$ into a $*$-Banach algebra (we denote them as $A_1$ and $A_2$). The obvious morphism $A_1\to A_2$, $a\mapsto a$ is bounded by definition.

Should it also be completely bounded? If not, are there any criteria to know when they are.

To this end, it could also be supposed that $A$ is unital and $\|1\|_i=1$.