The continuity of multiplication for two topologies associated with a semifinite trace on von Neumann algebra

(Lobachevskii Journal of Mathematics, Vol.14, pp.17-24)

Let ${\cal M}$ be a semifinite von Neumann algebra in a Hilbert space $\cal H$ and $\tau$ be a normal faithful semifinite trace on ${\cal M}$. Let ${{\cal M}}^{{\rm pr}}$ denote the set of all projections in ${\cal M}$, $e$ denote the unit of ${\cal M}$, and $\|\cdot\|$ denote the $C^*$-norm on ${\cal M}$. The set of all $\tau$-measurable operators $\widetilde {{\cal M}}$ with sum and product defined as the respective closures of the usual sum and product, is a *-algebra. The sets $$ U({\varepsilon, \delta})\!=\!\{x \in \widetilde {{\cal M}}\!: \|xp\|\le \varepsilon {\rm~and~} \tau(e-p)\le \delta {\rm~for~some~} p \in {{\cal M}}^{{\rm pr}} \}, \; \varepsilon\!>\!0, \, \delta \!>\!0, $$ form a base at $0$ for a metrizable vector topology $t_{\tau}$ on $\widetilde {{\cal M}}$, called {\it the measure topology}. Equipped with this topology, $\widetilde {{\cal M}}$ is a complete topological *-algebra. We will write $x_i \buildrel{\tau}\over \longrightarrow x$ in case a net $\{x_i{\}}_{i \in I} \subset \widetilde {{\cal M}}$ converges to $x \in \widetilde {{\cal M}}$ for the measure topology on $\widetilde {{\cal M}}$. By definition, a net $\{x_i{\}}_{i \in I} \subset \widetilde {{\cal M}}$ {\it converges $\tau$-locally to} $x \in \widetilde {{\cal M}}$ (notation: $x_i \buildrel{\tau l}\over \longrightarrow x$) if $x_ip \buildrel{\tau}\over \longrightarrow xp$ for all $p \in {{\cal M}}^{{\rm pr}}$, $\tau(p)<\infty$; and a net $\{x_i{\}}_{i \in I} \subset \widetilde {{\cal M}}$ {\it converges weak $\tau$-locally to} $x \in \widetilde {{\cal M}}$ (notation: $x_i \buildrel{w \tau l}\over \longrightarrow x$) if $px_ip \buildrel{\tau}\over \longrightarrow pxp$ for all $p \in {{\cal M}}^{{\rm pr}}$, $\tau(p)<\infty$. \medskip {\bf Theorem 1.} {\it Let $x_i,x \in \widetilde {{\cal M}}$. {\bf 1.} If $x_i \buildrel{\tau l}\over \longrightarrow x $, then $x_iy \buildrel{\tau l}\over \longrightarrow xy$ and $yx_i \buildrel{\tau l}\over \longrightarrow yx$ for every fixed $y \in {\widetilde {{\cal M}}}$. {\bf 2.} If $x_i \buildrel{w \tau l}\over \longrightarrow x $, then $x_iy \buildrel{w \tau l}\over \longrightarrow xy$ and $yx_i \buildrel{w \tau l}\over \longrightarrow yx$ for every fixed $y \in {\widetilde {{\cal M}}}$. } \medskip {\bf Theorem 2}. {\it If $\{x_i{\}}_{i \in I} \subset \widetilde {{\cal M}}$ is bounded in measure and if $x_i \buildrel{\tau l}\over \longrightarrow x \in \widetilde {{\cal M}}$, then $x_iy \buildrel{\tau}\over \longrightarrow xy$ for all $\tau$-compact $y \in {\widetilde {{\cal M}}}$. } \medskip {\bf Theorem 3}. {\it Let $x,y,x_i,y_i \in \widetilde {{\cal M}}$ and let a set $\{x_i{\}}_{i \in I} $ be bounded in measure. If $x_i \buildrel{\tau l}\over \longrightarrow x$ and $y_i \buildrel{\tau l}\over \longrightarrow y$, then $x_iy_i \buildrel{\tau l}\over \longrightarrow xy$. } If ${\cal M}$ is abelian, then the weak $\tau$-local and $\tau$-local convergencies on $\widetilde {{\cal M}}$ coincides with the familiar convergence locally in measure. If $\tau(e)=\infty$, then the boundedness condition cannot be omitted in Theorem~2. If ${\cal M}$ is ${\cal B}({\cal H})$ with standard trace, then Theorem~2 for sequences is a "Basic lemma"of the theory of projection methods: {\it If $y$ is compact and $x_n \to x$ strongly, then $x_ny \to xy$ uniformly, i.e. $\|x_ny - xy\| \to 0$ ${\rm ~as~} n \to \infty$.} Theorem 3 means that the mapping $$ (x,y)\mapsto xy : \; ({\cal B}({\cal H})_1 \times {\cal B}({\cal H}) \to {\cal B}({\cal H})) $$ is strong-operator continuous (${\cal B}({\cal H})_1$ denotes the unit ball of ${\cal B}({\cal H})$).



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