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.

  • $\begingroup$ Are you assuming $f$ is normal? $\endgroup$ – Yemon Choi May 29 '12 at 0:14
  • $\begingroup$ Any ${\rm II_1}$ factor has a unique trace, and it is indeed normal. $\endgroup$ – Jesse Peterson May 29 '12 at 20:30

Some hints:

  • By replacing $H$ by $H\otimes\ell^2$, which doesn't change separability, you may suppose that every normal state $f$ is a vector state $x\mapsto (x\xi|\xi)$.
  • Then $H_f$ is the completion of $M$ for $(x|y) = f(y^*x) = (y^*x\xi|\xi) = (x\xi|y\xi)$. So the map $x\mapsto x\xi$ extends to an isometry from $H_f$ into $H$. So $H_f$ is separable.

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


Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.