Let $\mathfrak{A}$ be a C${}^\*$ algebra and $\mathbb{R}\ni s \mapsto \alpha\_s$ a continuous family of its automorphisms. Is it true that


$$
\int d s \, f(s)\, \alpha\_s(A)
$$


is well defined as a Bochner integral for any $A\in\mathfrak{A}$ and $f \in L^1(\mathbb{R})$.



Integration in C^* algebra
Yes.


$\alpha\_s(A)$ is a continuous bounded function.


The function
$f(s) \alpha\_s(A)$ is measurable and because of


$$\int\_\mathbb{R} \|f(s) \alpha\_s(A)\| ds \le \int\_\mathbb{R} |f(s)| ds\, \|A\| < \infty$$


in $L^1$.


Quite elementary, look in the book "Serge Lang, Real and Functional Analysis", Chapter Integration for related things.


