Sums of commutators in non-commutative Banach function spaces

Abstract

Let M be a II ∞-factor and denote by τ its normal faithful semi-finite trace. For any rearrangement invariant Köthe function space X on [0,+∞[, let X( M, τ) be the associated non-commutative Banach function space. This paper is concerned with ideals in M of the form I X ( M, τ)= M∩ X( M, τ) that are contained in L p ( M, τ) for some p>0. It is proved that an element T in I X ( M, τ) is a finite sum of commutators of the form [ A, B] with A∈ I X ( M, τ) and B∈ M if and only if the function t→ 1 t ∫ |λ|>λ t(T) λ dν T(λ) belongs to X, where ν T is the Brown spectral measure of T and t→ λ t ( T) is the non-increasing rearrangement of the function λ→| λ| with respect to ν T . This extends to general Banach function spaces a result obtained by Kalton for quasi-Banach ideals of compact operators and implies that the Dixmier's trace of a quasi-nilpotent element in L 1,∞( M, τ) is always zero.