Algebraic and order properties of a block projection operator on the algebra of measurable operators
https://doi.org/10.26907/2541-7746.2025.4.641-654
Abstract
Let 𝜏 be a faithful normal semifinite trace on a von Neumann algebra ℳ. The block projection operator 𝒫̃𝑛 (𝑛 ≥ 2) on the *-algebra 𝑆(ℳ, 𝜏 ) of all 𝜏 -measurable operators is investigated. It is shown that 𝑓(𝒫̃𝑛(𝐴)) ≥ 𝒫̃𝑛(𝑓(𝐴)) for any operator monotone function 𝑓 on R+ and 𝐴 ∈ 𝑆(ℳ, 𝜏 )+. For an operator convex function 𝑓 on R+, we have 𝑓(𝒫̃𝑛(𝐴)) ≤ 𝒫̃𝑛(𝑓(𝐴)) for 𝐴 ∈ 𝑆(ℳ, 𝜏 )+. Conditions are established under which 𝒫̃𝑛(𝐴) belongs to the class 𝑆0(ℳ, 𝜏 ) of 𝜏 -compact operators, to the class 𝐹(ℳ, 𝜏 ) of elementary operators, to the classes 𝐿𝑝(ℳ, 𝜏 ) of operators 𝜏 -integrable with 𝑝-th power, or to the ℳ algebra itself. If 𝐴, 𝐵 ∈ 𝑆(ℳ, 𝜏 ) and 𝒫̃𝑛(𝐵) is a left (right) inverse for the operator 𝐴, then 𝒫̃𝑛(𝐵) is also a left (respectively, right) inverse for the operator 𝒫̃𝑛(𝐴).
Supporting agencies: This study was supported by the Development Program of the Research and Educational Mathematical Center of the Volga Region Federal District (agreement no. 075-02-2025-1725/1) and by the Development Program of the Regional Research and Educational Mathematical Center “Azov-Black Sea Mathematical Center” (agreement no. 075-02-2025-1543 dated February 27, 2025)
About the Authors
M. F. Darwish Taleb
Kazan Federal University
Russian Federation
Russian Federation
Mohammed Firas Darwish Taleb, Postgraduate Student
M. A. Muratov
V.I. Vernadsky Crimean Federal University
Russian Federation
Russian Federation
Mustafa A. Muratov, Dr. Sci. (Physics and Mathematics), Full Professor
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For citations:
Darwish Taleb M.F., Muratov M.A. Algebraic and order properties of a block projection operator on the algebra of measurable operators. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki. 2025;167(4):641-654. (In Russ.) https://doi.org/10.26907/2541-7746.2025.4.641-654
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