Geodesic
A block projection operator in the algebra of measurable operators
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 10 (2023), pp. 77-82.

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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)”.
Keywords: Hilbert space, von Neumann algebra, trace, measurable operator, block projection operator. 
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     title = {A block projection operator in the algebra of measurable operators},
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A. M. Bikchentaev. A block projection operator in the algebra of measurable operators. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 10 (2023), pp. 77-82. http://geodesic.mathdoc.fr/item/IVM_2023_10_a6/

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