A block projection operator in the algebra of measurable operators

Let $\tau$ be a faithful normal semifinite trace on a von Neumann algebra $\mathcal{M}$. We investigate the block projection operator $\mathcal{P}_n$ $(n\ge 2)$ in the ${}^*$-algebra $S(\mathcal{M}, \tau )$ of all $\tau$-measurable operators. We show that $A \leq n\mathcal{P}_n(A)$ for any operator $A\in S(\mathcal{M}, \tau)^+$. If an operator $A\in S(\mathcal{M}, \tau)^+$ is invertible in $S(\mathcal{M}, \tau)$ then $\mathcal{P}_n(A)$ is invertible in $S(\mathcal{M}, \tau)$. Consider $A=A^*\in S(\mathcal{M},\tau)$. Then (i) if $\mathcal{P}_n(A)\leq A$ $($or if $\mathcal{P}_n(A)\geq A)$ then $\mathcal{P}_n(A)= A$; (ii) $\mathcal{P}_n(A)= A$ if and only if $P_kA= AP_k$ for all $ k=1, \ldots, n$; (iii) if $A, \mathcal{P}_n(A)\in \mathcal{M}$ are projections then $\mathcal{P}_n(A)= A$. We obtain 4 corollaries. We also refined and reinforced one example from the paper “A. Bikchentaev, F. Sukochev, Inequalities for the block projection operators, J. Funct. Anal. 280 (7), article 108851, 18 p. (2021)”.