This question is closely related to this previous question.
I have an AFD factor $\mathcal{N} \subset B(\mathcal{H})$. There is a group action (for simplicity we take the group to be $\mathbb{Z}_2$) on $B(\mathcal{H})$ given by an automorphism $\theta = \mathrm{Ad}_{\Theta}$ with $\Theta^2 = \mathbb{I}$. It is further assumed that $\mathcal{N}$ is invariant under this group action; $\theta(\mathcal{N}) = \mathcal{N}$.
Question : Is $\mathcal{N}$ the weak closure of an increasing family of full matrix algebras $\{\mathcal{M}_i\}$ such that $\theta(\mathcal{M}_i) = \mathcal{M}_i$ for each $i$?
Context : The reason I ask this question is the same as in the previous question. In my application $\theta$ is a $\mathbb{Z}_2$-grading, so $\mathcal{N}$ is a super von Neumann algebra. I want to construct a conditional expectation onto the super-commutant of $\mathcal{N}$. I can construct such a conditional expectation if the answer to the above question is affirmative.
Thoughts so far : I describe here a reason why any naive attempt at constructing symmetric full matrix algebras from given full matrix algebras is likely to fail.
Consider a spin chain, i.e. to each 'site' $n \in \mathbb{Z}$ there is associated a copy of the full matrix algebra $\mathcal{A}_n \simeq \mathcal{M}_{2 \times 2}$. The algebra of local observables $\mathcal{A}$ is the norm closure of the direct limit of the family of their finite tensor products.
for finite $I \subset \mathbb{Z}$ denote by $\mathcal{A}_I = \otimes_{n \in I} \mathcal{A}_n$
A nearest-neighbour quantum cellular automaton (n.n. QCA) is an automorphism $\theta$ of $\mathcal{A}$ such that $\theta(\mathcal{A}_{n}) \subset \mathcal{A}_{\{n-1, n, n+1\}}$ for all sites $n$.
Let $\theta$ be a n.n. QCA such that $\theta^2 = \mathrm{id}$, i.e. $\theta$ is a $\mathbb{Z}_2$ group action.
To such a $\theta$ one can associate an index in $H^3(\mathbb{Z}_2, U(1))$ (I will not describe here how this is done, but see this paper). It is known that there are $\theta$ with non-trivial index.
On the other hand, if near some site there were $\theta$-invariant full matrix algebras that together generate an algebra containing at least two neighbouring $\mathcal{A}_n$'s, then the cohomology index must be trivial.
This indicates that for a $\theta$ with nontrivial index, if there are any invariant full matrix algebras, they must be very 'spread out'. This seems to destroy any hope of constructing invariant matrix algebras out of the given local matrix algebras.