Given two von-Neumann algebra factors $\mathcal M,\mathcal N$, is $\mathcal M\cap\mathcal N$ a factor?


And how about the intersection of infinitely many factors?


**Notes:**


* I know that the intersection is a von-Neumann algebra. (This is immediate from the definition of a von-Neumann algebra as a SOT-closed algebra with adjoints and 1.)
* I know that this does not hold for *Type I* factors. (See comments [here](https://mathoverflow.net/questions/442906/intersection-of-finitely-many-type-i-von-neumann-algebra-factors).)



Intersection of von-Neumann algebra factors
The answer is no. There are subfactors $N\subset M$ with finite Jones index $[M:N]$ with $N^{\prime}\cap M=\mathbb{C}\oplus \mathbb{C}$. For example, consider a type $II\_1$ factor $P$ and let $\alpha$ be an outer automorphism of $P$. Put $M= P\otimes M\_2(C)$ and $N$ be the algebra of all diagonal matrices $(x,\alpha(x)),$ for $x\in P$.


