Dixmier traces as singular symmetric functionals and applications to measurable operators.
| dc.contributor.author | Lord, S. | |
| dc.contributor.author | Sedaev, A. | |
| dc.contributor.author | Sukochev, F. | |
| dc.date.issued | 2005 | |
| dc.description | Copyright © 2005 Elsevier Inc. All rights reserved. | |
| dc.description.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 operator ideal ℒ<sup>(1,∞)</sup> (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<sup>(1,∞)</sup>, i.e. those on which an arbitrary Connes-Dixmier trace yields the same value. In the special case, when the operator ideal ℒ<sup>(1,∞)</sup> is considered on a type I infinite factor, a bounded operator x belongs to ℒ<sup>(1,∞)</sup> if and only if the sequence of singular numbers {s<inf>n</inf> (x)}<inf>n≥1</inf> (in the descending order and counting the multiplicities) satisfies ∥x∥<inf>(1,∞)</inf>:=sup<inf>N≥1</inf> 1/ Log(1+N) ∑<inf>n=1</inf><sup>N</sup> s<inf>n</inf> (x) < ∞. In this case, our characterization amounts to saying that a positive element x ∈ ℒ<sup>(1,∞)</sup> is measurable if and only if lim<inf>N→∞</inf> 1/Log N ∑<inf>n=1</inf><sup>N</sup> s<inf>n</inf> (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<sup>(1,∞)</sup>/ℒ<inf>0</inf> <sup>(1∞)</sup>, where the space L<inf>0</inf><sup>(1,∞)</sup> is the closure of all finite rank operators in ℒ<sup>(1,∞)</sup> in the norm ∥.∥<inf>(1,∞)</inf>. © 2005 Elsevier Inc. All rights reserved. | |
| dc.description.statementofresponsibility | Steven Lord, Aleksandr Sedaev and Fyodor Sukochev | |
| dc.description.uri | http://www.elsevier.com/wps/find/journaldescription.cws_home/622879/description#description | |
| dc.identifier.citation | Journal of Functional Analysis, 2005; 224(1):72-106 | |
| dc.identifier.doi | 10.1016/j.jfa.2005.01.002 | |
| dc.identifier.issn | 0022-1236 | |
| dc.identifier.issn | 1096-0783 | |
| dc.identifier.orcid | Lord, S. [0000-0002-6142-5358] | |
| dc.identifier.uri | http://hdl.handle.net/2440/43260 | |
| dc.language.iso | en | |
| dc.publisher | Academic Press Inc Elsevier Science | |
| dc.source.uri | https://doi.org/10.1016/j.jfa.2005.01.002 | |
| dc.subject | Non-normal (Dixmier) traces | |
| dc.subject | Singular symmetric functionals | |
| dc.subject | Banach limits | |
| dc.subject | Marcinkiewiczspaces | |
| dc.subject | Non-commutative geometry | |
| dc.title | Dixmier traces as singular symmetric functionals and applications to measurable operators. | |
| dc.type | Journal article | |
| pubs.publication-status | Published |