Differences of idempotents in $C^*$-algebras and the quantum Hall effect, II. Unbounded idempotents

Let the von Neumann algebra ${\mathcal M}$ of operators act in the Hilbert space $\mathcal{H}$, $\tau$ – exact normal semifinite trace on $\mathcal{M}$, $S(\mathcal{M}, \tau )$ – ${}^*$-algebra of $\tau$-measurable operators and $ L_1(\mathcal{M},\tau)$ is the Banach space of $\tau$-integrable operators. If $P, Q \in S(\mathcal{M}, \tau )^{\text{id}}$ and $P-Q\in L_1(\mathcal{M},\tau)$, then $\tau (P-Q)\in \mathbb{R}$. In particular, if $A=A^3\in L_1(\mathcal{M}, \tau )$, then $\tau (A)\in \mathbb{R}$. Let $A, B \in S(\mathcal{M}, \tau)$ be tripopotents. If $A-B\in L_1(\mathcal{M}, \tau )$ and $A+B\in \mathcal{M}$, then $\tau (A-B)\in \mathbb{R}$. Let $P, Q \in S(\mathcal{M}, \tau )^{\text{id)}}$ with $P-Q\in L_1(\mathcal{M},\tau)$ and $P Q \in \mathcal{M}$. Then for all $n\in \mathbb{N}$ we have $(P-Q)^{2n+1}\in L_1(\mathcal{M},\tau)$ and $\tau ((P-Q)^{2n+1})=\tau (P-Q)\in \mathbb{R}$.