I have a pure state $\omega$ on a quasilocal algebra $\mathcal{A}$ on a 2d lattice $\Gamma = \mathbb{Z}^2$ with a $\mathbb{C}^d$ vector space on each site. Let there be a unitary symmetry action $U_g(x) \in \mathcal{B}(\mathbb{C}^d)$ with $g \in G$, a finite group. $\omega$ is invariant under a symmetry automorphism $\alpha_g(A):= (\otimes_{x \in \Lambda} U_g(x)) A (\otimes_{x \in \Lambda} U_g^*(x))$ for all $A \in \mathcal{A}_{\Lambda}$ where $\Lambda$ is a finite subset of $\Gamma$. i.e, $\omega\circ \alpha_g = \omega$.
Now let's say I have $L$ as the left half of $\Gamma$ after cutting it in 2 parts and an automorphism $\alpha^L_g(A):= (\otimes_{x \in \Lambda \cap L} U_g(x)) A (\otimes_{x \in \Lambda \cap L} U_g^*(x))$ for all $A \in \mathcal{A}_{\Lambda}$ where $\Lambda$ is a finite subset of $\Gamma$.
Consider $(\pi, \mathcal{H})$ as the GNS representation of $\omega$. Is $\alpha^L_g$ implementable on the GNS Hilbert space of $\omega$ with a unitary? i.e, does there exist a unitary $U \in \mathcal{B}(\mathcal{H})$ such that $$\mathrm{Ad}[U] \circ \pi(A) = \pi \circ \alpha_g^L(A)$$ for all $A \in \mathcal{A}$?
Is $\alpha_g^L$ weakly continuous? I.e, If I have $A_n{\rightarrow} A$ in WOT then does $\alpha_L^g(A_n)$ converge in WOT?
EDIT: Some more information about $\omega$ in case it's relevant. It's a gapped ground state of a topological order, and its cone vN algebra $\pi(\mathcal{A}_{\Lambda})''$, where $\Lambda$ is a cone (including a half plane), is a type $II_{\infty}$ factor. It is also expected to satisfy the approximate-split property: If there are two cones $\Lambda_1, \Lambda_2$ such that $\Lambda_2$ is "sufficiently inside" $\Lambda_1$, then there exists a type $I$ factor $\mathcal{N}$ with $\pi(\mathcal{A}_{\Lambda_1})'' \subset \mathcal{N} \subset \pi(\mathcal{A}_{\Lambda_2})''$.