I will write the definition of the Arveson spectrum, but eventually me question is a very specific case.
Let $A$ be a $C^*$ algebra, let $G$ be a locally compact Abelian group and let $\alpha: G\to Aut(A)$ be a point norm continuous action.
For $f\in L^1(G)$ and $x\in A$ define $\alpha_f(x)=\int_{G} f(t)\alpha_t(x)dt$. The integral is a well-defined element of $A$.
We have: $\alpha_{f*g}=\alpha_f\circ \alpha_g$ for all $f,g\in L^1(G)$.
For $x\in A$, define $sp_{\alpha}(x)=\{\gamma\in \hat{G}: \forall f\in L^1(G), \alpha_f(x)=0\to \hat{f}(\gamma)=0\}$.
The Arveson spectrum of $\alpha$ is:
$sp(\alpha)=\{\gamma\in \hat{G}:$ for any open neighborhood $U$ of $\gamma$, $\exists 0\neq x\in A$ s.t. $sp_{\alpha}(x)\subseteq U\}$.
One can show that if $G$ is a compact group then:
$sp(\alpha)=\{\gamma\in \hat{G}: \exists 0\neq x\in A$ with $\alpha_t(x)=\gamma(t)x$ for all $t\in G\}$.
If $G$ is locally compact Abelian, then $\gamma\in \hat{G}$ if and only if there exists a sequence $\{x_n\}$ in the unit ball of $A$ s.t. $||\alpha_t(x_n)-\gamma(t)x_n||\to 0$ uniformly in $t$ on compact subsets of $G$.
Claim: If $\alpha:\Bbb{Z}\to Aut(A)$, then $sp(\alpha)=\sigma(\alpha_1)$, if we regard $\alpha_1$ as an element of $B(A)$.
I've succeeded to show the inclusion $sp(\alpha)\subseteq \sigma(\alpha)$. It follows from the identification of $\hat{\Bbb{Z}}$ with $\Bbb{T}$, the fact that compact subsets of $\Bbb{Z}$ are just finite subsets, so can be replaced with a generating set, i.e. $1$, and finally that approximate eigenvalues are in the spectrum of a bounded operator.
I don't know how to show the converse inclusion.
Moreover, I don't have any motivation for these definitions.
So, if you have any explanation it is not less important from my question above.
Thank you for any help!