Consider a general spacetime containing an observer, and let $\mathcal{A}_{\text{obs}$ denote the algebra of observables available to the observer. It has been proposed that the Hartle-Hawking "no boundary" state $\Psi_{HH}$ of the universe can be interpreted as a universal state of maximum entropy for $\mathcal{A}_{\text{obs}}$. Under this assumption, one can suggest a general definition of the entropy $S(\Psi)$ of any state $\Psi$ of the observer algebra, as the negative of the relative entropy of $\Psi$ with respect to $\Psi_{HH}$:
$$ S(\Psi) = -S(\Psi|\Psi_{HH}) $$
It has been argued heuristically that generically $\Psi_{HH}$ will not be a normalizable state in the Hilbert space describing the observer's observations but may be a "normal weight" on the observer's von Neumann algebra $\mathcal{A}_{\text{obs}}$. It is then speculated that under this maximum entropy hypothesis for $\Psi_{HH}$, the observer algebra will always be of Type II (either $\text{II}_1$ or $\text{II}_\infty$ depending on whether $\Psi_{HH}$ is normalizable or not).
Here are some questions I have:
Under what conditions on a spacetime and observer can the no-boundary state $\Psi_{HH}$ be rigorously defined as a normal weight on the observer algebra $\mathcal{A}_{\text{obs}}$?
Assuming $\Psi_{HH}$ exists as a normal weight, is the formula $S(\Psi) = -S(\Psi|\Psi_{HH})$ mathematically well-defined and physically sensible as a definition of entropy for arbitrary states $\Psi$?
Does the existence of $\Psi_{HH}$ as a faithful normal weight on $\mathcal{A}_{\text{obs}}$ imply that $\mathcal{A}_{\text{obs}}$ is necessarily a Type II von Neumann algebra? Can the $\text{II}_1$ vs $\text{II}_\infty$ distinction be related to normalizability of $\Psi_{HH}$ as conjectured?