# Strong operator convergence of support in non-commutative $L^p$ spaces

Let $$M$$ be a von Neumann algebras with normal faithful semifinite trace $$\tau.$$ Let $$L^p(M)$$be the associated non-commutative $$L^p$$-space. Suppose, $$x\geq 0$$ be an element of $$L^p(M)$$ and $$0\leq x_n\leq x$$ such that $$x_n$$ converges to $$x$$ in measure. Is it true that $$\text{support of}\ x_n$$ converges to support of $$x$$ in strong operator topology or weak operator topology or $$\sigma$$-strong topology?

• What is "convergence in measure" in a noncommutative $L^p$ space? – Nik Weaver May 16 at 14:17
• @Nik. Take the characteristic function $1_{(\lambda,\infty)}.$ We say $x_n\to x$ in measure if $1_{(\lambda,\infty)}(|x-x_n|)$ converges to $0$ as $n\to\infty$ for all $\lambda>0.$ – Samya Kumar Ray May 17 at 4:48