Thanks Eleni, Ray and John for the last nice posts.
Peter Morgan raises an extremely good issue, with both a technical and a conceptual side. I refer here to his post above, without trying to repeat here his points, since these are several, interconnected, and nicely expressed by Peter.
First, a technical point. It is true that Tomita theory wants a von Neumann algebra, and therefore the corresponding norm; but this is given for free by the state over the C* algebra. The reason is that such a state is enough for the GNS construction, which provides the representation, the Hilbert space, and therefore the von Neumann algebra structure. Thus, C* algebra and (appropriate) state are enough.
This does not answer Peter's questions, however; it only moves them one step back, because all the physical questions he poses about the norm, and in particular the meaning of its its probabilistic interpretation in a timeless context, can just be reformulated for the state itself.
The question, I think, raises deep issues. I'll try to answer is steps. First, regarding the operational meaning of probability. I do not want to enter the debate about probability here. I only state my preferences, without arguments. The definition of probability that makes sense to me is the one of de Finetti, subjective. Probability is something about my expectations, and the corresponding mathematical theory expresses the proper way of adjusting expectations on the basis of observations, where "proper" refers to the use of Bayes theorem, which follows from the logic structure of probabilities (expectations) themselves. We make (finite) sequences of observations, and readjust our previous hypotheses on the basis of outcomes.
This answers the question about operationalism, but not the question about the possibility of doing so in a timeless context. This is the difficult part of the question. The question is about the rationalization of experiencial temporality in a physical world. I am not sure physics in the strict sense is capable of answering it. The question pertains to that part of science that studies ourselves and our capacity to gather and elaborate informationa and conceive thoughts and representations of the world. What I am saying is that I think that timeless quantum gravity is equally blind to the temporal aspect of consciousnes as Newtonian mechanics is. What basic physics must provide, however, is a context within which it is possible to have the basic ingredients in terms of which a science of complex systems might make sense of processes of organizing information. So, here is the story: In a timeless world, a small subsystem (us) whose interaction with the rest of the universe is limited to a very small number of variables, and therefore who has no access to the exact state of the rest of the universe (that is, it has the same state for many different states of the universe), can be correlated with the rest of the world in such a way to have an imprecise information about the rest of the system (a way to express these notions precisely using Shannon information theory is in my work on relational quantum theory); then with respect to this subsystem a Tomita flow is defined; and this flow itself is the physical underpinning of the perception of the flow of time, whatever this perception is.
Maybe I should not have entered this domain, which is slipery. I think that it is better to keep as separated as possible physics and the science about cognitive capacities. Otherwise we make confusion. As an analogy: we understand the water molecules, from there the hydrodynamica behavior of liquid water, from there the floating of boats, including the boat on which we stand when we collect water from the sea on order to study its molecular strucure. But trying to write a theory of molecular structure of water thinking that within the theory we may directly see out floating boat is the bad procedure. Here the analogy is: water molecules/timeless mechanics, floating/tomita flow, collecting water from the boat/operational definition of probability.
Hope this was not too confused.
Carlo Rovelli