I have a C* algebra A, a function $f(x)\in A$ and an analytic function $F:\mathbb{C}\rightarrow\mathbb{C}$. I would like to know what condition must have $F$ such that $F(f)\in A$.
The idea is the following: F must be analytic on a neighborhood of $Im(f)$ but I've been trying to see how to prove it or some reference about it, but I haven't found nothing...
You can define a holomorphic functional calculus on any complex Banach algebra with identity. If $F$ is analytic in a neighbourhood $U$ of the spectrum $\sigma(f)$, take a finite collection of simple closed contours $\Gamma$ such that every point around which $\Gamma$ has nonzero winding number is in $U$, and $\Gamma$ has winding number $1$ around every point of $\sigma(f)$. Then define $$ F(f) = \frac{1}{2\pi i} \oint_\Gamma F(z) (z - f)^{-1} \; dz$$