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 P_n (n ≥ 2) in the ^∗-algebra S(M, τ ) of all τ -measurable operators. We show that A ≤ nP_n(A) for any operator A ∈ S(M, τ )⁺. If an operator A ∈ S(M, τ )+ is invertible in S(M, τ ) then P_n(A) is invertible in S(M, τ ). Consider A = A^∗ ∈ S(M, τ ). Then (i)   if P_n(A) ≤ A (or if P_n(A) ≥ A) then P_n(A) = A; (ii) P_n(A) = A if and only if P_kA = AP_k 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)”.