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Zbl 1077.46055
Bikchentaev, A. M.
The continuity of multiplication for two topologies associated with a semifinite trace on von Neumann algebra. (English)
[J] Lobachevskii J. Math. 14, 17-24, electronic only (2004). ISSN 1995-0802; ISSN 1818-9962

Let $M$ be a semifinite von Neumann algebra equipped with a faithful normal semifinite trace $\tau$, and let $\widetilde{M}$ denote the topological $*$-algebra of $\tau$-measurable operators. We say that a net ${x_i}\subset \widetilde{M}$ converges $\tau$-locally (resp., weak $\tau$-locally) to $x\in \widetilde{M}$ if, for each projection $p\in M$ of finite trace, $x_ip \to xp$ (resp., $px_ip\to pxp$) in measure. It is shown that: multiplication by a fixed $y\in \widetilde{M}$, either on the left or on the right, is continuous for both $\tau$-local and weak $\tau$-local topologies; multiplication is jointly continuous for $\tau$-local topology if the net on the left is bounded in measure; multiplication on the right by a $\tau$-compact operator takes $\tau$-locally convergent sequences into measure convergent ones.
[Stanisław Goldstein (MR 2005a:46130)]

MSC 2000:: *46L51 Noncommutative measure and integration

Keywords: von Neumann algebra; measure topology; convergence in measure; convergence locally in measure; compact operator; semifinite trace

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