Suppose we have type $II_1$ factor $\mathcal{M}$ acting on separable Hilbert space $H$. Consider a faithful tracial state $f=tr$ (we know that such object exists) and produce $H_f$ as a Hilbert space obtained by the GNS construction from the state $f$. Does it follows that $H_f$ is separable Hilbert space? Going into details in GNS contruction suggest that this may fail to happen while $H_f$ is completion of $\mathcal{M}$but we take a completion with respect to smaller norm.

Some hints:
This doesn't use any special about $M$ or $f$. 

