# type II$_1$ subalgebra of type III$_1$ factor

Let $$M$$ be a type III$$_1$$ factor and $$N$$ be the type II$$_1$$ subalgebra of $$M$$.

What is the type of $$N'\cap M$$? Can it be any type?

Such a relative commutant $$N' \cap M$$ can even be any factor. Choose any type III$$_1$$ factor $$P$$ with a type II$$_1$$ subfactor $$N \subset P$$ with trivial relative commutant $$N' \cap P = \mathbb{C} 1$$. When $$Q$$ is an arbitrary factor, we get that $$P \overline{\otimes} Q$$ is a type III$$_1$$ factor that contains the type II$$_1$$ subfactor $$N \otimes 1$$ with relative commutant $$1 \otimes Q$$.
• When $P_1$ is any type III$_1$ factor, one can realize $P_1$ as the crossed product of a II$_\infty$ factor $N_1$ and a trace scaling action of $\mathbb{R}$. Then $N_1$ is a type II$_\infty$ subfactor of $P_1$ with trivial relative commutant. Now take a projection $p \in N_1$ that is finite in $N_1$. Then $N = p N_1 p$ is a type II$_1$ subfactor of $P = p P_1 p$ with trivial relative commutant. The conclusion is that any type III$_1$ factor has a type II$_1$ subfactor with trivial relative commutant. Feb 21 at 10:09