Let $\mathcal M$ be a finite von Neumann algebra and $\mathcal E:\mathcal M\to\mathcal N$ be the conditional expectation operator onto a von Neumann subalgebra. Suppose that we know $\|ex_ne\|_\infty\leq c$ for a fixed sequence $(x_n)\in \mathcal N$ and a projection $e\in\mathcal M$.
Is it true that $\sup_{n\geq 1}\|ax_na\|_\infty<\infty$ where $a:=\mathcal E(e)$ ?
This fails even in finite dimension. Let $M=M_2(\mathbb C)$, $N=\mathbb C\oplus\mathbb C$, and $\mathcal E$ the canonical pinching, that is $$ \mathcal E\bigg(\begin{bmatrix} a&b\\c&d\end{bmatrix}\bigg)=\begin{bmatrix} a&0\\0&d\end{bmatrix}. $$ Let $$ x_n=\begin{bmatrix} n&0\\0&-n\end{bmatrix}\in N,\qquad\qquad e=\frac12\begin{bmatrix} 1&1\\1&1\end{bmatrix}. $$ Then $ex_ne=0$ for all $n$, but since $\mathcal E(e)=\frac12\,I_2$, $$ \|ax_na\|=\frac14\,\bigg\|\begin{bmatrix} n&0\\0&-n\end{bmatrix}\bigg\|=\frac n4. $$