Commutator structure of operator ideals

By K. Dykema, T. Figiel, G. Weiss, Mariusz Wodzickid. 2004

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Dimensions and singular traces for spectral triples, with applications to fractals (2002, background): “Recently, we became aware of a related result contained in [ 11 ], where non-positive trace functionals on B(H) are studied.”
Dixmier traces and some applications in non-commutative geometry (2006, background): “The proof given in [35] is based on deep results in [32] (see also [54], [55]) concerning the structure of the commutator space [I, J ] spanned by all the commutators [T, S], T ∈ I, S ∈ J (here, I and J are ideals of compact operators in L (H )).”
An infinite dimensional Schur-Horn theorem and majorization theory with applications to operator ideals (2007, background): “ectively, ξa := h 1 n Xn 1 ξji and ξa∞ := h 1 n X∞ j=n+1 ξji 36 VICTOR KAFTAL AND GARY WEISS are essential for the study of commutator spaces (i.e., commutator ideals) and hence traces on ideals (see [7] and [16] -[21].) If I is an ideal, then the arithmetic mean ideals aI and Ia, called the pre-ari”