relative commutant of type III1 factor

Question

Let $M$ be a type III$_1$ factor and $N$ be the non-trivial ($N\neq \Bbb C 1$)semi-finite von Neumann subalgebra of $M$.

What is the type of relative commutant $N'\cap M$? Is it semi-finite?

Answer 1

There are examples when the relative commutant is not semi-finite. If $B$ is a type $\mathrm{III}_1$ factor, then $M=B\overline\otimes\mathbb{B}(H)\cong B$ for any separable Hilbert space $H$. Now if you take $N=\mathbb C\otimes B(H)$, then $N$ is semi-finite with relative commutant $N^\prime\cap M=B$, which is type $\mathrm{III}_1$ and hence not semi-finite.