Let $\mathfrak{A}$ be a C${}^*$ algebra and $\mathbb{R}\ni s \mapsto \alpha_s$ a 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 $f \in L^1(\mathbb{R})$.

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 $f \in L^1(\mathbb{R})$.