Let $ \mathcal{H} $ be a not-necessarily-separable Hilbert space. Let $ G $ be a locally compact Hausdorff group. It is easy to see that if $ U: G \to \mathbb{U}(\mathcal{H}) $ is a norm-continuous homomorphism from $ G $ to the group of unitary operators on $ \mathcal{H} $, then we can define a strongly continuous action $ \alpha $ of $ G $ on the $ C^{\ast} $-algebra $ \mathbb{K}(\mathcal{H}) $ of compact operators on $ \mathcal{H} $ by $$ \forall g \in G: \qquad \alpha_{g} \stackrel{\text{df}}{=} \left\{ \begin{matrix} \mathbb{K}(\mathcal{H}) & \to & \mathbb{K}(\mathcal{H}) \\ T & \mapsto & U(g) \circ T \circ U(g^{-1}) \end{matrix} \right\}. $$

Question.If $ \alpha $ is astrongly continuousaction of $ G $ on $ \mathbb{K}(\mathcal{H}) $, does it necessarily arise from anorm-continuoushomomorphism from $ G $ to $ \mathbb{U}(\mathcal{H}) $ as in the manner described above?

I am pretty sure that the answer to this is known already, but I am frustrated by my inability to locate a reference.

Thank you so much for your assistance!

**Clarification**

To avoid any confusion, I wish to clarify that a group acts on a $ C^{\ast} $-algebra by $ \ast $-automorphisms.

from the contextof the question that the OP is asking about homomorphisms $\alpha: G \to {\rm Aut}{\mathbb K}({\mathcal H})$ where the automorphisms are in the category of ${\rm C}^*$-algebras -- which makes them isometric, $*$-preserving, non-degenerate, etc etc. $\endgroup$8more comments