0
$\begingroup$

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.

  • 0
    Actually nothing. After understanding this and some results that were obtainable afterwards, I now understand that raising to a given real power is a continuous operation on any bounded set within the positive cone, hence the entire positive cone. (My original question was about the continuity of this operation, and it's only definable on the positive cone, so this is as much as I could hope for.)2012-08-19

1 Answers 1

4

For $A\mapsto A^n$ to be continuous, it is enough for multiplication to be continuous. Since multiplication is continuous in any normed algebra (it's an obvious consequence of submultiplicativity of the norm), the map is continuous in any normed algebra.

To show that use induction with respect to $n$.

  • 0
    Actually, by translating between the abstract proof we've given so far and what it means when it's norms and not topologies. Just write $x^n-y^n$ as the telescoping sum $x^n-x^{n-1}y+x^{n-1}y...$ We just have different intuitions I guess, but I wanted to do something like the high school commutative case with that usual expansion, and when I couldn't I thought that only some minor modification would be needed, when it turns out that you actually just have to change strategies.2012-08-19