The topologies of local convergence in measure on the algebras of measurable operators

Given a von Neumann algebra ${\mathcal M}$ of operators on a Hilbert space ${\mathcal H}$ and a faithful normal semifinite trace $\tau$ on ${\mathcal M}$, denote by $S({\mathcal M}, \tau )$ the $*$-algebra of $\tau$-measurable operators. We obtain a sufficient condition for the positivity of an hermitian operator in $S({\mathcal M}, \tau )$ in terms of the topology $t_{\tau l}$ of $\tau$-local convergence in measure. We prove that the $*$-ideal ${\mathcal F}({\mathcal M}, \tau )$ of elementary operators is \hbox{$t_{ \tau l}$-dense} in $S({\mathcal M}, \tau )$. If $t_{ \tau}$ is locally convex then so is $t_{ \tau l}$; if $t_{ \tau l}$ is locally convex then so is the topology $t_{w \tau l}$ of weakly $\tau$-local convergence in measure. We propose some method for constructing $F$-normed ideal spaces, henceforth $F$-NIPs, on $({\mathcal M}, \tau )$ starting from a prescribed $F$-NIP and preserving completeness, local convexity, local boundedness, or normability whenever present in the original. Given two \hbox{$F$-NIPs ${\mathcal X}$} and ${\mathcal Y}$ on $({\mathcal M}, \tau )$, suppose that $A{\mathcal X}\subseteq {\mathcal Y}$ for some operator $A \in S({\mathcal M}, \tau )$. Then the multiplier ${\mathbf M}_A X=AX$ acting as ${\mathbf M}_A : {\mathcal X}\to {\mathcal Y}$ is continuous. In particular, for ${\mathcal X}\subseteq {\mathcal Y}$ the natural embedding of ${\mathcal X}$ into ${\mathcal Y}$ is continuous. We inspect the properties of decreasing sequences of $F$-NIPs on $({\mathcal M},\tau )$.