Completely equivalent operator norms on $*$-Banach algebras.

A priori, it does not make sense to talk about complete boundedness, since there are no specified operator space structure on $A_1$ and $A_2$.

In general, an infinite-dimensional Banach space can carry many incomparable operator space structure. Most prominently, there is the minimal and the maximal operator space structure (see Chapter 3 in the book of Gilles Pisier (see here). These two almost never the same.

There are criteria (also due to Pisier) which ensure that certain bounded maps between $C^*$-algebras are automatically completely bounded. This is related to the notion of length of a $C^*$-algebra. This is also explained in his book.