For the sake of this question, lets call a factor $M$ *approximately semifinite* if there exists an increasing net of semifinite subfactors $M_i$, $i\in J$, with conditional expectations $E_i:M\to M_i$ such that

- $E_i\circ E_j = E_i$ if $i\ge j$,
- For all $\varphi\in M_*$ it holds that $\lim_i ||\varphi-\varphi\circ E_i|| =0$.

Clearly, all semifimite factors are approximately semifinite and I know of several results where it is shown that certain type $III$ factors are approximately semifinite. For instance, all type $III_0$ factors with separable predual are approximately semifinite, see [1, Prop. 8.3]. Also, every ITPFI factor $M=\bigotimes_{n=1}^\infty (M_{k_n},\varphi_n)$ is approximately semifinite (with $E_i: M \to \bigotimes_{n=1}^i M_{k_n}$, $i\in J=\mathbb N$, being the slice map $E_i(x\otimes y)= (\otimes_{n=i+1}^\infty \varphi_n)(y)\cdot x$ relative to $M=(\bigotimes_{n=1}^i M_{k_n})\otimes (\bigotimes_{n=i+1}^\infty M_{k_n})$. Since all non-ITPFI hyperfinite factors are of type $III_0$, this shows that every hyperfinite factor is approximately semifinite.

My question is: How large is the class of approximately semifinite factors really? Are factors (with separable predual) known that are not approximately semifinite?

[1] *Haagerup, Uffe; Størmer, Erling*, **Equivalence of normal states on von Neumann algebras and the flow of weights**, Adv. Math. 83, No. 2, 180-262 (1990).