good night... I was looking into the Pedersen Book, $C^{*}$-Algebras and their automorphism groups, and found the definition of analytic elements $x\in A$, where $(A,\alpha)$ is a $C^{*}-$dynamical system.

We say that an element $x\in A$ is analytic for $\alpha$ if the function $t\mapsto \alpha_{t}(x)$ has an extension, necessarily unique, to an analytic (entire) vector-valued function $\zeta\rightarrow \alpha_{\zeta}(x)$, $\zeta\in\mathbb{C}$. If $x\in A$ then $$x_n=\sqrt{\frac{n}{\pi}}\int_{\mathbb{R}}\alpha_t(x)exp(-nt^{2})dt$$ is analytic for $\alpha$.

And it's not clear for me that $x_n$ is well defined, i.e., why $x_n\in A$? and how is the integral defined:

$[1]$ The integration theory for vector-valued functions in a general Banach space, with Riemann sums, or

$[2]$ The Lebesgue integral for Banach spaces is the Bochner integral.

$\alpha_t(x_n)$ extends to $$\alpha_{\zeta}(x_n)=\sqrt{\frac{n}{\pi}}\int_{\mathbb{R}}\alpha_t(x)exp(-n(t-\zeta)^{2})dt$$

Analyticity of $\alpha_{\zeta}(x_n)$ is automatic because $\alpha_t(x)exp(-n(t-\zeta)^{2})$ is continuous and the Fundamental Theorem of Calculus?

I hope you answer me, i really thanks full. Do you recommend some bibliography?

Thanks so much.