Throughout, let us assume that $G$ denotes a group with the so-called infinite conjugacy class (ICC) property; i.e., the property that that the conjugacy class of any non-trivial element is infinite. A subgroup $H$ of $G$ is residual if there exists a subset $S \subseteq G \setminus H$ and elements $g_1 , g_2 \in G$ such that $G \setminus H = S \cup g_1^{-1} S g_1$ and $S, g_2^{-1}S g_2$, and $g_2 S g_2^{-1}$ are disjoint subsets of $G \setminus H$.
Given a group $G$, let $L(G)$ denote its group von Neumann algebra, which is generated by the left regular representation $\lambda : G \to U(\ell^2(G))$. Since $\delta_e$ is a cyclic and separating for $L(G)$, $x \mapsto x \delta_e$ is a linear isometry with dense range. In particular, using this fact, given any $x \in L(G)$, we can write $x = \sum_{g \in G} x_g \lambda_g$ for some scalars $x_g \in \mathbb{C}$ (Fourier coefficients) satisfying $\sum_{g \in G} |x_g|^2 < \infty$ and $\lambda_g$ denotes the unitary induced by $g \in G$ from $G$'s left regular representation. The von Neumann algebra $L(G)$ comes with a unique trace $\tau : L(G) \to \mathbb{C}$ given by $\tau (x) = \langle x \delta_{e} , \delta_e \rangle_{\ell^2(G)}$, making $L(G)$ is a type $II_1$ factor (because $G$ is ICC). Given a subgroup $H \le G$, let $E_{H} : L(G) \to L(H)$ be defined by $E_{H}(\sum_{g \in G} x_g \lambda_g) = \sum_{g \in H} x_g \lambda_g$ is a (the) conditional expectation of $L(G)$ onto $L(H)$.
Let $M$ be a $II_1$ factor von Neumann algebra with unique trace $\tau : M \to \mathbb{C}$. We can use $\tau$ to get a norm on $M$ defined via $||x||_2 = \sqrt{\tau (x^*x)}$. A sequence of elements $(x_n)_{n \in \mathbb{N}} \subseteq M$ bounded with respect to the trace norm (2-norm) is a central sequence if $||x_n y - y x_n||_2 \to 0$ as $n \to \infty$ for every $y \in M$.
I am trying to prove the following:
If $H$ is a residual subgroup of $G$, then any central sequence in $L(G)$ is equivalent to a central sequence whose elements lie in $L(H)$.
Given a central sequence $(x_n)_{n \in \mathbb{N}}$ in $L(G)$, it's pretty clear that $(E_{H}(x_n))_{n \in \mathbb{N}}$ is the desired central sequence. It's easy to show that $(E_{H}(x_n))_{n \in \mathbb{N}}$ is $2$-norm bounded, because $E_{H}$ is norm one projection, and it is easy to show that $(E_{H}(x_n))_{n \in \mathbb{N}}$ asymptotically commutes with everything in $L(H)$ in the $2$-norm. However, I am having trouble showing it commutes with everything in $L(G)$ and it's equivalent to $(x_n)_{n \in \mathbb{N}}$.