On Idempotent $\tau$-Measurable Operators Affiliated to a von Neumann Algebra

Let $\tau$ be a faithful normal semifinite trace on a von Neumann algebra $\mathscr M$, let $p$, $0<p<\infty$, be a number, and let $L_p(\mathscr M,\tau)$ be the space of operators whose $p$th power is integrable (with respect to $\tau$). Let $P$ and $Q$ be $\tau$-measurable idempotents, and let $A\equiv P-Q$. In this case, 1) if $A\ge 0$, then $A$ is a projection and $QA=AQ=0$; 2) if $P$ is quasinormal, then $P$ is a projection; 3) if $Q\in\mathscr M$ and $A\in L_p(\mathscr M, \tau)$, then $A^2\in L_p(\mathscr M,\tau)$. Let $n$ be a positive integer, $n>2$, and $A=A^n\in\mathscr M$. In this case, 1) if $A\ne 0$, then the values of the nonincreasing rearrangement $\mu_t(A)$ belong to the set $\{0\}\cup[\|A^{n-2}\|^{-1},\|A\|]$ for all $t>0$; 2) either $\mu_t(A)\ge 1$ for all $t>0$ or there is a $t_0>0$ such that $\mu_t(A)=0$ for all $t>t_0$. For every $\tau$-measurable idempotent $Q$, there is a unique rank projection $P\in\mathscr M$ with $QP=P$, $PQ=Q$, and $P\mathscr M=Q\mathscr M$. There is a unique decomposition $Q=P+Z$, where $Z^2=0$, $ZP=0$, and $PZ=Z$. Here, if $Q\in L_p(\mathscr M,\tau)$, then $P$ is integrable, and $\tau(Q)=\tau(P)$ for $p=1$. If $A\in L_1(\mathscr M,\tau)$ and if $A=A^3$ and $A-A^2\in\mathscr M$, then $\tau(A)\in\mathbb R$.