Let $A,B,C:L^2(\mathbb{T}^n)\to L^2(\mathbb{T}^n)$. Which conditions on $B$ I need to impose in order to say that $\Vert ABC\Vert\le \Vert B\Vert \Vert AC\Vert$?
I thought maybe to impose conditions on $A$ as well. For instance If we had a condition on $A$ such that $\Vert ABC\Vert\le \Vert BAC\Vert$ it would have been sufficient for the inequality to hold.
You cannot get away with imposing conditions only on $B$, unless the condition is $B=0$ or $B=I$. Indeed, if $B\ne0$ and $B\ne I$, take linearly independent $x,y$ with $y=Bx$. Let $C$ be the rank-one operator with $Cx=x$, and $A$ a rank-one operator with $Ax=0$ and $Ay=x$. Then $ABCx=x$, so $\|ABC\|\geq1$, while $AC=0$.
If you allow conditions on $A$, then a sufficient condition is that $AB=BA$.