There are some theorems that say in a unital C* algebra $A$ when one can deduce that the functional calculus of a continuous function f is continuous as map from some subset of $A$ to $A$. In the proof of one of these theorems it assumes that $A \mapsto A^n$ is continuous, and that $A \mapsto p(A, A^*)$ is continuous for any polynomial in $z, \bar z$. I actually suspect that $A \mapsto A^n$ is continuous in a more general setting where the latter statement wouldn't make sense, namely Banach algebras or even normed algebras. Please let me know why this is true, or at any greater or lesser level of generality, as long as it contains the C* algebra case. Thanks.
Incidentally, I am aware that raising to powers can be not continuous for powers more than $1$. If the integer powers give rise to continuous maps as I've conjectured above, I'd appreciate an example that illuminates why real positive powers can be so different.