Trace inequalities for measurable operators affiliated to a von Neumann algebra

Abstract

Let φ be a trace on von Neumann algebra M, A, B ϵ M and ||B|| < 1, [A, B] = AB—BA. Then φ(|[A,B]|) ≤ 2 φ(|A|). Let τ be a faithful normal semifinite trace on M, S(M, τ) be the *-algebra of all τ-measurable operators. If A ϵ L₂(M, τ) and Re A = λ|A| with λ ϵ {—1,1}, then A = λ|A|. An operator A ϵ L₂(M, τ) is Hermitian if and only if τ(A²) = τ(A*A). Let positive operators A, B ϵ S(M, τ) be invertible in S(M, τ) and Y := (A^-1 — B^-1)(A — B). If Y, A^1/2YA^-1/2 ϵ L₁(M, τ), then τ(Y) ≤ 0. Let an operator A ϵ S(M, τ) be hyponormal and A = B + iC be its Cartesian decomposition. If 1) BC ϵ L₁(M, τ), оr 2) C = C³ ϵ M and [B, C] ϵ L₁(M, τ), then A is normal.