Dixmier traces as singular symmetric functionals and applications to measurable operators

We unify various constructions and contribute to the theory of singular symmetric functionals on Marcinkiewicz function/operator spaces. This affords a new approach to the non-normal Dixmier and Connes-Dixmier traces (introduced by Dixmier and adapted to non-commutative geometry by Connes) living on a general Marcinkiewicz space associated with an arbitrary semifinite von Neumann algebra. The corollaries to our approach, stated in terms of the operator ideal L(1,∞) (which is a special example of an operator Marcinkiewicz space), are: (i) a new characterization of the set of all positive measurable operators from L(1,∞), i.e. those on which an arbitrary Connes-Dixmier trace yields the same value. In the special case, when the operator ideal L(1,∞) is considered on a type I infinite factor, a bounded operator x belongs to L(1,∞) if and only if the sequence of singular numbers {sn(x)}n⩾1 (in the descending order and counting the multiplicities) satisfies ∥x∥(1,∞)≔supN⩾11Log(1+N)∑n=1Nsn(x)<∞. In this case, our characterization amounts to saying that a positive element x∈L(1,∞) is measurable if and only if limN→∞1LogN∑n=1Nsn(x) exists; (ii) the set of Dixmier traces and the set of Connes-Dixmier traces are norming sets (up to equivalence) for the space L(1,∞)/L0(1∞), where the space L0(1,∞) is the closure of all finite rank operators in L(1,∞) in the norm ∥.∥(1,∞).