Let \(\mathcal{M}\) be a von Neumann algebra of operators acting on a Hilbert space \(\mathcal{H}\) equipped with a faithful normal semifinite trace \(\tau\). It has become almost classical to define notions like the \(*\)-algebra \(S(\mathcal{M},\tau)\) of \(\tau\)-measurable operators on \(\mathcal{H}\), its ideals \(S_0(\mathcal{M},\tau)\) (respectively \(\mathcal{F}(\mathcal{M},\tau)\)) of \(\tau\)-compact (respectively elementary) operators. For \(0<p<\infty\), counterparts of the classical Lebesgue spaces \(L^p(\mathcal{M},\tau)\) are defined in a natural way with an F-norm \(\|\cdot\|_p\) (which is a norm if \(p\ge1\)). In the case \(\mathcal{M}=\mathcal{B}(\mathcal{H})\) (the bounded operators on \(\mathcal{H}\)), we have that \(S(\mathcal{M},\tau)=\mathcal{M}\), \(S_0(\mathcal{M},\tau)=\) the compact operators on \(\mathcal{H}\), \(\mathcal{F}(\mathcal{M},\tau)=\) the finite rank operators on \(\mathcal{H}\). Most results below are new even in this special case.
The author continues the study of the topology \(t_\tau\) (respectively \(t_{\tau l}\), \(t_{w\tau l}\)) of convergence (respectively local, weak local convergence) in measure. Zero neighborhoods of \(t_{\tau l}\) are of the form \[\mathcal{V}(\varepsilon,\delta,P)=\{X\in S(\mathcal{M},\tau) : \exists Q\in\mathcal{M}\text{ a projection s.t. } Q\le P,\ \|XQ\|\le\varepsilon, \ \tau(P-Q)\le\delta\}\] where \(P\) is a projection of \(\mathcal{M}\), \(\varepsilon>0\), \(\delta >0\). If \(P\) is replaced by the unit \(I\) of \(\mathcal{M}\) (respectively if \(\|XQ\|\) is replaced by \(\|QXQ\|\)), then one obtains the zero neighborhoods of \(t_\tau\) (respectively \(t_{w\tau l}\)). The three versions coincide if \(\tau(I)<\infty\). Sometimes the local versions are more appropriate than \(t_\tau\). For example, in the classical space \(L^1(\mu)\) with a \(\sigma\)-finite \(\mu\), a sequence in \(L^1(\mu)\) converges in norm if and only if it converges in the weak and in the local measure topology.
A \(*\)-linear subspace \(\mathcal{X}\subset S(\mathcal{M},\tau)\) with an F-norm \(\|\cdot\|_\mathcal{X}\) is called an F-normed space (F-NIP) on \((\mathcal{M},\tau)\) if the involution on it is isometric for \(\|\cdot\|_\mathcal{X}\) and if, given \(A\in\mathcal{X}\), \(B\in S(\mathcal{M},\tau)\), the inequality \(|B|\le|A|\) implies that \(B\in\mathcal{X}\) and \(\|B\|_\mathcal{X}\le\|A\|_\mathcal{X}\). The inclusion of such an \(\mathcal{X}\) in \(S(\mathcal{M},\tau)\) is known to be \(t_{w\tau l}\)-continuous.
We list some results of the paper.

1.

There is a sufficient condition for a selfadjoint \(Y\in S(\mathcal{M},\tau)\) to be positive: if \(X\in S(\mathcal{M},\tau)\) is such that \(X^n\to0\) in \(t_{\tau l}\) and \(X^*YX\le Y\), then \(Y\ge0\).

2.

The \(*\)-ideal \(\mathcal{F}(\mathcal{M},\tau)\) (and a fortiori \(L^p(\mathcal{M},\tau)\)) is \(t_{\tau l}\)-dense in \(S(\mathcal{M},\tau)\).

3.

If \(t_\tau\) is locally convex (on \(S(\mathcal{M},\tau)\)), then so is \(t_{\tau l}\); if \(t_{\tau l}\) is locally convex, then so is \(t_{w\tau l}\).

4.

Given an invertible selfadjoint element \(A\) in the commutator of \(S(\mathcal{M},\tau)\), the author associates to an F-NIP \(\mathcal{X}\) another F-NIP, a kind of weighted F-NIP, \(\mathcal{X}(A)\), which inherits properties from \(\mathcal{X}\) (if \(\mathcal{X}\) has them) like local convexity, completeness and others.

5.

If \(\mathcal{X}\subset\mathcal{Y}\) are two F-NIPs (on \(S(\mathcal{M},\tau)\)), then the inclusion of \(\mathcal{X}\) into \(\mathcal{Y}\) is \(t_{w\tau l}\)-continuous.

6.

The intersection \(\mathcal{X}=\bigcap\mathcal{X}_n\) of a sequence \((\mathcal{X}_n)\) of F-NIPs on \(S(\mathcal{M},\tau)\) becomes naturally an F-NIP. Some results on such an intersection are obtained. For example, if \(\mathcal{Y}\) is a locally bounded F-NIP, then \(\mathcal{X}\subset\mathcal{Y}\) if and only if \(\mathcal{X}\subset\mathcal{Y}_n\) for some \(n\in\mathbb{N}\).