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

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