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:
$$
 P\_C^\perp P\_A \perp P\_C^\perp P\_B?
$$



Collection of projection operators in finite dimension and algebraic techinques
I don't know about question 2, but question 1 can indeed be answered using a general result about the projection lattice $P$ (ordered by $p\leq q\Leftrightarrow p=pq$) of a von Neumann algebra $A$.



> 
> $Q=\{q\in P:pa=qa\}$ is a complete sublattice of $P$, for any $a\in A$ and $p\in P$
> 
> 
> 


Proof: Let $[b]$ denote the range projection of any $b\in A$. If $R\subseteq Q$ then, for all $q\in R$, $[pa]=[qa]\leq q$ so $r=\bigwedge R$ satisfies $[qa]\leq r\leq q$ and hence $ra=rqa=r[qa]qa=[qa]qa=qa=pa$, i.e. $r\in Q$. So $Q$ is closed under taking infimums and, as $pa=qa\Leftrightarrow p^\perp a=q^\perp a$ (where $p^\perp=1-p$), the same applies to $Q^\perp=\{q^\perp:q\in Q\}$. But $p\leq q\Leftrightarrow q^\perp\leq p^\perp$ so this is saying $Q$ is closed under taking supremums, i.e. $Q$ is a complete sublattice of $P$. $\Box$


In fact, the above proof works more generally for any Baer \*-ring $A$ (see Berberian's book "Baer \*-rings"), or even Rickart \*-ring $A$ (where $Q$ is a complete sublattice of $P$ means that $Q$ is closed under infimums and supremums whenever they exist).


