Disclaimer: This is a record of results.
Given a C*-algebra $\mathcal{A}\subseteq\mathcal{B}(\mathcal{H})$.
Consider a Hamiltonian dynamics: $$H:\mathcal{D}\to\mathcal{H}:\quad\tau^t[A]:=e^{itH}Ae^{-itH}$$
Clearly, for bounded Hamiltonians: $$H\in\mathcal{B}(\mathcal{H}):\quad\tau^t[A]\stackrel{t\to0}{\to}A$$
Trivially, it holds also for: $$\mathcal{A}=\mathbb{C}:\quad\tau^t[\lambda1]\stackrel{t\to0}{\to}\lambda1$$
What nontrivial examples are there: $$\mathcal{A}\neq\mathbb{C}:\quad\tau^t[A]\stackrel{t\to0}{\to}A\quad(H\notin\mathcal{B}(\mathcal{H}))$$
Given the Fock space $\mathcal{F}(\mathcal{h})$.
Consider the second quantization: $$\Gamma(e^{ith})=e^{it\mathrm{d}\Gamma(h)}$$
It acts continuous on the CAR-algebra: $$\quad\tau^t[a(\eta)]=a(e^{ith}\eta)\stackrel{t\to0}{\to}a(\eta)$$ (Remember that this won't work for the Weyl algebra.)