Let $M\_1$ and $M\_2$ be von Neumann algebras acting on Hilbert spaces $H\_1,H\_2$ and consider $M=M\_1\overline\otimes M\_2$ acting on $H\_1\otimes H\_2$. Let $K$ be an $M$-invariant subspace (so that $P\_K\in M\_1'\overline\otimes M\_2'$).
Assume that $PMP = B(K)$.


Consider the von Neumann algebras $R\_1 = P M\_1\otimes 1 P $ and $R\_2 = P1\otimes M\_2 P$.
Then $R\_1\vee R\_2=B(K)$ and $R\_1\subset R\_2'$, so both $R\_i$, $i=1,2$, are factors.
Therefore $R\_1\subset R\_2'$ is an irreducible subfactor inclusion.


My question is: Do I overlook something that already implies that $R\_1 = R\_2'$? If not, which irreducible subfactor inclusions can arise this way?


A more general question would be if normal \*homomorphisms preserve relative commutants. This has surely been answered but I could not find it.



Question on tensor product of von Neumann algebras and subfactors
I would say you get $R\_1 = R\_2'$, and also $R\_1 = B(K')$ for some $K'$.


1. The projection $P$ is minimal in $M\_1' \mathbin{\bar\otimes} M\_2'$:
In general, if $M \subset B(H)$ is a von Neumann algebra and $P \in M'$ with range $K$, you have $(M P)' = P M' P$ on $K$.
(You can find this in "Takesaki I", Proposition II.3.10.)
From your assumption $M P = B(K)$, you get $P M' P = \mathbb C P$.
2. The algebras $M\_i$ can be assumed to be type I factors:
First look at the type (I, II, III) decomposition of $M\_i'$.
Use the fact that algebras of type II and III cannot contain minimal projections, and that the type I component of the tensor product $M'$ only comes from the type I components of $M\_i'$.
This reduces the situation to the type I case.
Then you can further reduce it to the factor case by looking at the central support of $P$.
3. Now we know that $H\_i = H\_{i 1} \otimes H\_{i 2}$ for $i = 1, 2$, with $M\_i = B(H\_{i 1}) \mathbin{\bar\otimes} \mathbb C$, $M\_i' = \mathbb C \mathbin{\bar\otimes} B(H\_{i 2})$, and $P$ is a rank $1$ projection on $H\_{1 2} \otimes H\_{2 2}$.
Again from point 1 you get
$$
R\_2' = P B(H\_{1 1} \otimes H\_{1 2} \otimes H\_{2 2}) P = B(H\_{1 1}) P = R\_1.
$$


