Proposition 37.1.6.label Let $A$ be a $C^{*}$-algebra, $H_{1}, H_{2}$ be a complex Hilbert spaces, $\pi_{1}: A \to B(H_{1})$ and $\pi_{2}: A \to B(H_{2})$ be injective representations of $A$, and $U: H_{1} \to H_{2}$ be an unitary equivalence, then the mapping
\[\pi_{1}(A) \to \pi_{2}(A) \quad T \mapsto UTU^{-1}\]
is strong-operator and weak-operator continuous.
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