A priori, it does not make sense to talk about complete boundedness, since there are no specified operator space structure on A1 and A2.
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.
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