Tensor products of finite number of different objects are always well described in the literature. However, the situation of infinite tensor products seems to be much tougher. Even in the simplest case, the infinite tensor product of Hilbert spaces, we have to be patient to find a good references (apart from the original article by von Neumann). Can one explain or give the references to how the infinite tensor products of infinitely many (normal) states is defined. Is there an analogue to the "stabilizing sequence" and all but finite number of states are e.g. vector states?

If we consider an infinte tensor product of von Neumann algebras, it's predual seem to correspond to the space which consists of infinite tensor product of states, right?

Last question is a bit vague, but I want to ask it. Say that we have the predual of the infinte tensor product of the same von Neumann algebra $M$. We take the state $\omega \in M_{*}$, can we view $\omega^{\otimes_{\sigma-\text{weak}} n}$ for some $n \in \mathbb{N}$ an element of the predual of the infinite tensor product of $M$ ? When the limit of this expression makes sense I mean $\lim_{n \to \infty} \omega^{\otimes n}$, for instance when it is an element of the predual of the infinite tensor product of $M$ and how does it possibly look.

I'm studying "INFINITE TENSOR PRODUCTS OF VON NEUMANN ALGEBRAS" by Nakagami, so maybe by the end I will be able to answer my question, but still I will be grateful for any help.