Let $H$ be a (separable) Hilbert space and let $(\alpha_t)_{t\in \mathbb{R}}$ be a group of automorhisms of compacts $K(H)$. Assume moreover that this group is continuous, i.e. for each $T\in K(H)$ the map $\mathbb{R} \ni t \mapsto \alpha_t(T)\in K(H)$ is norm continuous.
It is known that automorphisms of $K(H)$ are implemented by unitaries, hence there exists a family $(U_t)_{t\in \mathbb{R}}$ of unitaries on $H$ such that $\alpha_t(T)=U_t T U_t^*$ for each $t,T$.
I have two questions:
1) Can we choose $U_t$ in such a way that $(U_t)_{t\in \mathbb{R}}$ is a group of operators, i.e. $U_{t+s}=U_t U_s$?
2) Assume that $(U_t)_{t\in \mathbb{R}}$ is a group of operators. Is it strongly continuous (for each $\xi\in H$ the map $\mathbb{R}\ni t\mapsto U_t \xi \in H$ is continuous) ?
By the result of von Neumann, if $(U_t)_{t\in \mathbb{R}}$ is a group and maps $\mathbb{R} \ni t \mapsto \langle \xi | U_t \eta \rangle\in \mathbb{C}$ are measurable then in fact $(U_t)_{t\in \mathbb{R}}$ is strongly continuous.
Alright, I have found an answer: we can always find a family of unitaries $(U_t)_{t\in \mathbb{R}}$ which will implement $(\alpha_t)_{t\in\mathbb{R}}$ and moreover will be a strongly continuous group. It is (a special case of) a main result in "Groups of Inner Automorphisms of von Neumann Algebras" - Robert R. Kallman, Journal of Functional Analysis 7, 43-60 (1971)