Conditional expectation with respect to von-Neumann subalgebras

This post is inspired by this paper by D. Shlyakhtenko ( arxiv link to similar result). While reading this paper, I came across some notation in formula in the statement of lemma 4 that I was unfamiliar with

\[f\left( \sum_{i=1}^{N+1} X^{(k)}_{i} : \left\{\sum_{i=1}^{N+1}X^{(r)}_{i}\right\}_{r\neq k}\right) = E_{W^{\star}\left(\left\{\sum_{i=1}^{N+1} X^{(r)}_{i}\right\}_{r=1}^{n}\right)}f\left(\sum_{i\neq j}X_{i}^{(k)} : \left\{\sum_{i \neq j} X_{i}^{(r)}\right\}_{r \neq k}, \left\{X_{j}^{r}\right\}_{r=1}^{n}\right)\]

In particular, what is this

\[E_{W^{\star}\left(\left\{\sum_{i=1}^{N+1} X^{(r)}_{i}\right\}_{r=1}^{n}\right)}\]

and what does this mean? My advisor told me that the $E$ represents a conditional expectation and the $W^\star$ represents a von-Neumann subalgebra. So, the goal of this post is to unpack this notation entirely.

Background

Let $(\Omega, \Sigma, \mathbb{P})$ be a probability space. Consider the commutative von-Neumann algebra of real valued random variables $A = L^{\infty}(\Omega, \mathbb{P})$ where we make the identification of $A \ni f \equiv M_{f} \in \mathcal{B}(L^2(\Omega, \mathbb{P}))$ where $M_{f}$ as as the multiplication operator acting by $L^2(\Omega, \mathbb{P}) \ni g \mapsto fg$.

In which case we will have the tracial state $\tau$ on $A$ being equal to simply the expectation $E[X] = \int_{\Omega} X(\omega) d\mathbb{P}(\omega)$. Note: we may sometmes replace $\mathbb{P}$ with $\mu$, but both indicate the same probability measure on $\Omega$.

Let $(X_{1}, \ldots, X_{n})$ and $(Y_{1}, \ldots, Y_{n})$ be $n$-tuples random variables. Let $P, Q \in A$ be polynomials in $X_{i}$s and $Y_{i}$s respectively. Then we have

\[E[P] = \int_{\Omega} P(X_{1}(\omega), \ldots, X_{n}(\omega))d\mathbb{P}(\omega)\]

and, similarly,

\[E[Q] = \int_{\Omega} P(Y_{1}(\omega),\ldots,Y_{n}(\omega))d\mathbb{P}(\omega)\]

Moreover, we say that the tuples of $X_{i}$s and $Y_{i}$s are independent if and only iff $E[PQ] = E[QP] = E[P]E[Q]$ whenever $P$ and $Q$ are polynomials in $X_{i}$s and $Y_{i}$s respectively.

Now, we define the space $L^{2}(A, E(\cdot))$.

The standard definition of conditional expectation takes place in the arena of some probability space $(\Omega, \Sigma, \mathcal{P})$ with two random variables $X$ and $Y$. Then, intuitively, we can construct a new random variable $Z = E[X|Y]$. This new random variable gives us the expectation of $X$ given $Y$ occurs. A simple example of this given on the wikipedia page for conditional expectation. However,