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A beginner mathmatician
A beginner mathmatician
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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$$\text{support of}\ x_n$ converges to support of $x$ in strong operator topology or weak operator topology or $\sigma$-strong topology?

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?

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?

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A beginner mathmatician
A beginner mathmatician
  • 1.9k
  • 6
  • 11

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?