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.)