A block projection operator in the algebra of measurable operators

Abstract

Let τ be a faithful normal semifinite trace on a von Neumann algebra M. We investigate the block projection operator Pn (n ≥ 2) in the ∗-algebra S(M, τ ) of all τ -measurable operators. We show that A ≤ nPn(A) for any operator A ∈ S(M, τ )+. If an operator A ∈ S(M, τ )+ is invertible in S(M, τ ) then Pn(A) is invertible in S(M, τ ). Consider A = A∗ ∈ S(M, τ ). Then (i)   if Pn(A) ≤ A (or if Pn(A) ≥ A) then Pn(A) = A; (ii) Pn(A) = A if and only if PkA = APk for all k = 1, . . . , n; (iii) if  A,   n(A) are projections then 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)”.