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?

  • $\begingroup$ What is "convergence in measure" in a noncommutative $L^p$ space? $\endgroup$ – Nik Weaver May 16 at 14:17
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    $\begingroup$ @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 $\lambda>0.$ $\endgroup$ – Samya Kumar Ray May 17 at 4:48

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