Analyticity of C*-algebra valued functions Let $\mathcal{A}$ be a unital C*-algebra and consider a function $f:\mathbb{C} \rightarrow \mathcal{A}$. What is an accessible tool to prove or disprove that $f$ is analytic, i.e. can be locally expanded in a power series of $z$?
Take as a concrete example $f_A(z) = \exp(zA-\overline{z}A^*)$ for some fixed $A\in \mathcal{A}$. Is $f_A$ an analytic function?
 A: The holomorphic functional calculus implies that any function that you would "expect" to be analytic is analytic and that the power series is exactly what it "should" be. For example, $z \mapsto e^z$ is analytic and has power series $\sum_{n=0}^\infty \frac{z^n}{n!}$.
However, we would not expect your example $f_A(z)=exp(z A - \bar z A^*)$ to be analytic. To see this, let's specialize to the case where $\mathcal A=\mathbb C$ and $A=1$. Then your function is $$f(z)=exp(2i \ Im \ z),$$ where $Im \ z$ is the imaginary part of $z$.  Power series should be in $z$, not in $Im \ z$ or $Re \ z$. Indeed, when we check whether the Cauchy-Riemann equations are satisfied, we see that the function $f$ is not analytic.
For details on the holomorphic functional calculus, you can take a look at Conway's A Course in Functional Analysis, Section VII.4.
If you are looking for conditions equivalent to analyticity, you can find a couple in Takesaki's Theory of Operator Algebras III, google books link. (There are surely better references available, but this is what I have by my side.)
