Algebraic and order properties of a block projection operator on the algebra of measurable operators

Let $\tau$ be a faithful normal semifinite trace on a von Neumann algebra $\mathcal{M}$. The block projection operator $\widetilde{\mathcal{P}}_n$ ($n \ge 2$) on the *-algebra $S(\mathcal{M},\tau)$ of all $\tau$-measurable operators is investigated. It is shown that ${f(\widetilde{\mathcal{P}}_n(A)) \ge \widetilde{\mathcal{P}}_n(f(A))}$ for any operator monotone function $f$ on $\mathbb{R}^+$ and ${A \in S(\mathcal{M},\tau)^+}$. For an operator convex function $f$ on $\mathbb{R}^+$, we have ${f(\widetilde{\mathcal{P}}_n(A)) \le \widetilde{\mathcal{P}}_n(f(A))}$ for ${A \in S(\mathcal{M},\tau)^+}$. Conditions are established under which $\widetilde{\mathcal{P}}_n(A)$ belongs to the class $S_0(\mathcal{M},\tau)$ of $\tau$-compact operators, to the class $ F(\mathcal{M},\tau)$ of elementary operators, to the classes $L_p(\mathcal{M},\tau)$ of operators $\tau$-integrable with $p$-th power, or to the $\mathcal{M}$ algebra itself. If ${A,B\in S(\mathcal{M},\tau)}$ and $\widetilde{\mathcal{P}}_n(B)$ is a left (right) inverse for the operator $A$, then $\widetilde{\mathcal{P}}_n(B)$ is also a left (respectively, right) inverse for the operator $\widetilde{\mathcal{P}}_n(A)$.