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.

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,