$\newcommand{\HH}{\mathrm{HH}}$Consider a general spacetime containing an observer, and let $\mathcal{A}_{\mathrm{obs}}$ 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 $\mathcal{A}_{\mathrm{obs}}$. 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 $\mathcal{A}_{\mathrm{obs}}$. 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:

1. 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 $\mathcal{A}_{\text{obs}}$?

2. 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 $\Psi$?

3. 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?