Sums of commutators in non-commutative Banach function spaces

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 IX(M,τ)=M∩X(M,τ) that are contained in Lp(M,τ) for some p>0. It is proved that an element T in IX(M,τ) is a finite sum of commutators of the form [A,B] with A∈IX(M,τ) and B∈M if and only if the function t→1t∫|λ|>λt(T)λ dνT(λ) belongs to X, where νT is the Brown spectral measure of T and t→λt(T) is the non-increasing rearrangement of the function λ→|λ| with respect to νT. This extends to general Banach function spaces a result obtained by Kalton for quasi-Banach ideals of compact operators and implies that the Dixmierʹs trace of a quasi-nilpotent element in L1,∞(M,τ) is always zero.