On Hyperfinite ness

Question

If $M\subset B(\mathcal{H})$ is hyperfinite type $\mathrm{II}_1$ factor, does it imply $PM$ is again hyperfinite type $\mathrm{II}_1$, where $P$ is a projection in $M'$.

Answer 1

It is known (see, for instance, Proposition 5.5.5 in Kadison Ringrose) that $MP\simeq MC_P$, where $C_P$ is the central carrier of $P$. As $C_P\in Z(M')=Z(M)=\mathbb C I$, we have $C_P=I$ and so $PM\simeq M$ via the explicit isomorphism $Pm\longmapsto m $.