# Projections in atomless von Neumann algebras

Let $$\varepsilon>0$$. If we consider a sequence $$\{f_n\}$$ in $$L_\infty(0,1)$$, then there exists a very small subset $$A$$ of $$(0,1)$$ with $$m(A)<\varepsilon$$ such that $$\|f_n \chi_A\|_\infty =\|f_n \|_\infty$$ for all $$n$$. My question is, do we have an analogue of this result in the von Neuamnn algebra setting? Precisely, let $$M$$ be an atomless von Neumann algebra and let $$\{x_n\}$$ be a sequence in $$M$$. Can we find a projection $$p$$ in $$M$$ which is very small (say, for a semifinite faithful normal weight, $$\omega(p)<\varepsilon$$) such that $$\|x_n p\|_\infty =\|x_n \|_\infty$$ for all $$n$$.

For the case of a semifinite von Neumann algebra, it is true because we may take $$p_n$$ to be very small such that $$\|x_n p_n\|_\infty =\|x_n\|_\infty$$ and let $$p:=\vee p_n$$. Since $$\tau(p)\le \sum_{n\ge 1}\tau(p_n)$$， we may choose suitable $$p_n$$ such that $$\tau(p)<\varepsilon$$. Moreover, $$\|x_n p\|_\infty =\|x_n \|_\infty$$ for all $$n$$. However, for the type III case, it seems to be rather difficult.