We obtain general theorems which enable the calculation of the Dixmier trace in terms of the asymptotics of the zeta function and of the trace of the heat semigroup. We prove our results in a general semi-finite von Neumann algebra. We find for p > 1 that the asymptotics of the zeta function determines an ideal strictly larger than L p,∞ on which the Dixmier trace may be defined. We also establish stronger versions of other results on Dixmier traces and zeta functions.

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 oper-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 {s n (x)} n 1 (in the descending order and counting the multiplicities) satisfies x (1,∞) := sup N 1 1 Log(1+N) N n=1 s n (x) < ∞. In this case, our characterization amounts to saying that a positive element x ∈ L (1,∞) is measurable if and only if lim N→∞ 1 Log N N n=1 s n (x) exists; (ii) the set of Dixmier traces and the * Corresponding author. Fax: +61 882012904. E-mail addresses: sed@vmail.ru (A. Sedaev), sukochev@infoeng.flinders.edu.au (F.A. Sukochev).

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For symmetric spaces of measurable functions on the real half-line, we study the problem of existence of positive linear functionals monotone with respect to the Hardy-Littlewood semi-ordering, the so-called symmetric functionals. Two new wide classes of symmetric spaces are constructed which are distinct from Marcinkiewicz spaces and for which the set of symmetric functionals is nonempty. We consider a new construction of singular symmetric functionals based on the translation-invariance of Banach limits defined on the space of bounded sequences. We prove the existence of Banach limits invariant under the action of the Hardy operator and all dilation operators. This result is used to establish the stability of the new construction of singular symmetric functionals for an important class of generating sequences.

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