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 $\|1\|\_i=1$.



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](http://books.google.de/books?id=0pKL-o7WUOAC)). 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.


