Dixmier traces as singular symmetric functionals and applications to measurable operators

Abstract

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).

Lidskii-Type Formulae for Dixmier Traces

“…Dixmier traces τ ω defined such ω's are termed Connes-Dixmier traces. We refer to [6,7,15] for discussion of their properties. Finally, various formulae of noncommutative geometry (in particular, those involving heat kernel estimates and generalised ζ-function) were established in [3,5,7] for yet a smaller subset of Connes-Dixmier traces, when the functional ω was assumed to be M -invariant.…”

“…Similarly, the definitions of Connes-Dixmier traces and M -invariant traces naturally extend to denote corresponding singular traces on Marcinkiewicz ideals M ψ (see [15]). Our main result for general Dixmier traces on ideals M ψ is given in Theorem 31 which asserts that for any Dixmier trace τ ω on M ψ with ψ ∈ Ψ satisfying condition (4) we have…”

The Dixmier trace and asymptotics of zeta functions

“…Notice that in this definition we have chosen ω to satisfy only the dilation invariance assumption even though Dixmier [20] originally imposed on ω the assumption of dilation and translation invariance (see [33] for more details).…”

A Lidskii type formula for Dixmier traces

“…Our results depend crucially on the recent characterization of positive measurable operators from L (1,∞) as those for which the limit lim t →∞ 1 log(1+t) t 0 µ s (T ) ds exists [11,Theorem 6.6], and the spectral characterization of sums of commutators in type II factors [8,9].…”