A Block Projection Operator in the Algebra of Measurable Operators

Abstract

Let $\tau$ be a faithful normal semifinite trace on a von Neumann algebra $\mathcal{M}$. The block projection operator ${{\mathcal{P}}_{n}}$ $(n \geqslant 2)$ in the *-algebra $S(\mathcal{M},\tau )$ of all $\tau$-measurable operators is investigated. It has been shown that $A \leqslant n{{\mathcal{P}}_{n}}(A)$ for any operator $A \in S{{(\mathcal{M},\tau )}^{ + }}$. If $A \in S{{(\mathcal{M},\tau )}^{ + }}$ is invertible in $S(\mathcal{M},\tau )$, then ${{\mathcal{P}}_{n}}(A)$ is invertible in $S(\mathcal{M},\tau )$. Let $A = A\text{*} \in S(\mathcal{M},\tau )$. Then, (i) if ${{\mathcal{P}}_{n}}(A) \leqslant A$ (or if ${{\mathcal{P}}_{n}}(A) \geqslant A$), then ${{\mathcal{P}}_{n}}(A) = A$, (ii) ${{\mathcal{P}}_{n}}(A) = A$ if and only if ${{P}_{k}}A = A{{P}_{k}}$ for all $k = 1, \ldots ,n$; and (iii) if $A,{{\mathcal{P}}_{n}}(A) \in \mathcal{M}$ are projections, then ${{\mathcal{P}}_{n}}(A) = A$. Four corollaries have been obtained. One example presented in paper (A. Bikchentaev and F. Sukochev, “Inequalities for the Block Projection Operators,” J. Funct. Anal. 280 (7), 108851 (2021)) has been refined and strengthened.