Trace inequalities for measurable operators affiliated to a von Neumann algebra

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 L 2 ( M , τ ) and Re A = λ| A | with λ ϵ {—1,1}, then A = λ| A |. An operator A ϵ L 2 ( M , τ ) is Hermitian if and only if τ ( A 2 ) = 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 1/2 YA -1/2 ϵ L 1 ( M , τ ), then τ ( Y ) ≤ 0. Let an operator A ϵ S ( M , τ ) be hyponormal and A = B + i C be its Cartesian decomposition. If 1) BC ϵ L 1 ( M , τ ), оr 2) C = C 3 ϵ M and [ B , C ] ϵ L 1 ( M , τ ), then A is normal.