Comparison of topologies on *-algebras of locally measurable operators

Details Comparison of topologies on *-algebras of locally measurable operators Comparison of topologies on *-algebras of locally measurable operators

2010-01-11

arxiv.org/abs/1001.1651v1

Mathematics

Operator Algebras

21 pages

Scientific paper

We consider the locally measure topology $t(\mathcal{M})$ on the *-algebra $LS(\mathcal{M})$ of all locally measurable operators affiliated with a von Neumann algebra $\mathcal{M}$. We prove that $t(\mathcal{M})$ coincides with the $(o)$-topology on $LS_h(\mathcal{M})=\{T\in LS(\mathcal{M}): T^*=T\}$ if and only if the algebra $\mathcal{M}$ is $\sigma$-finite and a finite algebra. We study relationships between the topology $t(\mathcal{M})$ and various topologies generated by faithful normal semifinite traces on $\mathcal{M}$.

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