Given an abelian C*-algebra $\mathcal{A}$, a state $\omega$, a strongly continuous group of *-automorphism $\{\tau_t : t \in \mathcal{R}\}$, and given a representation $ (\pi(\mathcal{A}), \mid \Omega \rangle)$, where $\pi: \mathcal{A} \to \mathcal{B}(H)$ and $H$ is a Hilbert space, such that $\forall a \in \mathcal{A}, \, \omega (a) = \langle \Omega \mid \pi(a) \Omega \rangle$, is it true that $\pi(\tau_t(a)) = U_t \pi(a) U_{t}^{\dagger} \iff \omega(\tau_t(a)) =\omega(a) \, \forall a \in \mathcal{A}, \, t \in \mathcal{R} $ ?

I was trivially thinking that, since the algebra is abelian, then

$\omega(\tau_t(a)) = \langle \Omega \mid U_t \pi(a) U_{t}^{\dagger} \Omega \rangle = \langle \Omega \mid U_t U_{t}^{\dagger}\pi(a) \Omega \rangle = \langle \Omega \mid \pi(a) \Omega \rangle = \omega(a)$,

but I am afraid I'm missing something, so any help is very welcome.

I have already posted this question on MathStackExchange, but without any luck: https://math.stackexchange.com/questions/4790179/covariant-representation-of-abelian-algebra