Sums of commutators in non-commutative Banach function spaces.
In [J. Reine Angew. Math. 504, 115--125 (1998; Zbl 0919.47022)], \textit{N. J. Kalton} gave a complete spectral characterization of \([I,B(H)]\) for a large class of two-sided ideals \(I\) of compact operators. Here, \([I,B(H)]\) denotes the linear span of commutators \([A,B]=AB-BA\) with \(A\in I\) and \(B\in B(H)\). The importance of the theorem stems from its applications to noncommutative geometry. Fack extends the result, replacing \(B(H)\) with a type \(II_\infty\) factor \(M\) and \(I\) with the set \(I_X(M,\tau)\) of elements of \(M\) belonging to a noncommutative Banach function space \(X(M,\tau)\) associated with a rearrangement invariant Köthe function space \(X\) on \([0,+\infty[\). The main theorem gives conditions under which \(T\) belongs to \(I_X(M,\tau)\) in terms of Brown's spectral measure of \(T\). The result implies that the Dixmier trace of a quasinilpotent element in \(L^{1,\infty}(M,\tau)\) is always zero. This very well written paper contains a wealth of material on noncommutative Banach function spaces.
scientific article; zbMATH DE number 2055848
scientific article