A block projection operator in the algebra of measurable operators

A. M. Bikchentaev

Izvestiâ vysših učebnyh zavedenij. Matematika, no. 10 (2023), pp. 77-82

Résumé

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)”.

Détail

Keywords: Hilbert space, von Neumann algebra, trace, measurable operator, block projection operator.