Consider a set of linearly independent vectors $\{x_1,\dots,x_n\}$ in some finite-dimensional Hilbert space $H$. For any subset $S \subset [n]$, let $P_S$ be the (orthogonal) projection (operator) onto the span of $x_S := \{x_i, \;i \in S\}$. Let us also write $P_j = P_{\{j\}}$.

We would like to study the collection of projections $\{P_S : \; S \subset [n]\}$. We also have some extra information which can be encoded in the form of a graph $G = ([n], E)$ such that for any $(i,j) \notin E$ \begin{align*} P_S^\perp P_i \perp P_S^\perp P_j, \quad S=[n]\setminus \{i,j\}. \end{align*} In other words, the residual errors after projecting $x_i$ on $x_S$ and $x_j$ on $x_S$ are orthogonal for any two nodes $i,j$ not connected with an edge.

My question is: Are there known algebraic techniques that help study these projections? Searching around, it seems that there is some connection to (finite-dimensional) von Neumann algebras, but I don't know much about them to see the link.

As a concrete question consider this: Fix $j \in [n]$ and $S \subset [n]\setminus\{j\}$ and consider $$ \mathcal{T}_j(S) := \{ T \subset [n]\setminus\{j\}:\; P_T P_j = P_S P_j\}. $$ I believe $\mathcal{T}_j(S)$ is a complete lattice (and the minimum and maximum elements can be read from the graph $G$ ...). Does this follow easily from a more general result?

EDIT: Concrete question 2: Consider $A,B,C \subset [n]$ such that $C$ separates $A$ and $B$ in graph $G$, i.e., there is no path in $G$ from $A$ to $B$ that does not share a node with $C$. Then, do we have: $$ P_C^\perp P_A \perp P_C^\perp P_B? $$