Commutator Structure of Operator Ideals1

Not much has been known about these spaces except for some special cases. It was known since 1953 that every operator A ∈ B(H) is the sum of two commutators of bounded linear operators [35], [16] while the commutator spaces of Schatten ideals Lp , p > 0, were the subject of several studies over the last 30 years, notably [49], [66], [3], [68], [67], [4], [38] and [18]. Aside from these efforts, the little that was known about the commutator spaces [I, J] did not even suffice for deciding whether equality [I, J] = [B(H), I J] held for general operator ideals. The study of this classical subject acquired a renewed sense of urgency since the introduction of cyclic cohomology by Alain Connes [20], [21], [22] in the early 1980’s. The work of the last author exhibited in 1991 a direct