Trace and Commutators of Measurable Operators Affiliated to a Von Neumann Algebra

Abstract

In this paper, we present new properties of the space L₁(M, τ) of integrable (with respect to the trace τ ) operators affiliated to a semifinite von Neumann algebra M. For self-adjoint τ-measurable operators A and B, we find sufficient conditions of the τ -integrability of the operator λI −AB and the real-valuedness of the trace τ (λI − AB), where λ ∈ ℝ. Under these conditions, [A,B] = AB − BA ∈ L₁(M, τ) and τ ([A,B]) = 0. For τ -measurable operators A and B = B², we find conditions that are sufficient for the validity of the relation τ ([A,B]) = 0. For an isometry U ∈ M and a nonnegative τ -measurable operator A, we prove that U − A ∈ L₁(M, τ) if and only if I − A, I − U ∈ L₁(M, τ). For a τ -measurable operator A, we present estimates of the trace of the autocommutator [A^∗,A]. Let self-adjoint τ -measurable operators X ≥ 0 and Y be such that [X^1/2, YX^1/2] ∈ L₁(M, τ). Then τ ([X^1/2, YX^1/2]) = it, where t ∈ ℝ and t = 0 for XY ∈ L₁(M, τ).