Dixmier traces as singular symmetric functionals and applications to measurable operators.
Date
2005
Authors
Lord, S.
Sedaev, A.
Sukochev, F.
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Journal article
Citation
Journal of Functional Analysis, 2005; 224(1):72-106
Statement of Responsibility
Steven Lord, Aleksandr Sedaev and Fyodor Sukochev
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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.
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Copyright © 2005 Elsevier Inc. All rights reserved.