Singular Symmetric Functionals

Dodds, P. G.; Pagter, B. de; Sedaev, A. A.; Semenov, E. M.; Sukochev, Fedor

doi:10.1023/b:joth.0000042448.87252.0f

Cited by 23 publications

(43 citation statements)

References 5 publications

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“…It is clear that (5) holds and, therefore, Marcinkiewicz space M ψ admits nonzero Dixmier traces (see [9][10][11]). Now we show that formula (1) holds for all Connes-Dixmier traces on M ψ .…”

Section: Lidskii Formula For Connes-dixmier Tracesmentioning

confidence: 99%

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Lidskii-Type Formulae for Dixmier Traces

Sedaev

Sukochev

Zanin

2010

Integr. Equ. Oper. Theory

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Section: Lidskii Formula For Connes-dixmier Tracesmentioning

confidence: 99%

“…We refer the reader to [9][10][11] for conditions which guarantee the additivity of τ ω . It is well known that τ ω is additive for any ω as above when…”

mentioning

confidence: 99%

Lidskii-Type Formulae for Dixmier Traces

Sedaev

Sukochev

Zanin

2010

Integr. Equ. Oper. Theory

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“…The established theory of Banach limits [10] and singular symmetric functionals on Marcinkiewicz spaces [4][5][6] can be applied to questions concerning the (Connes-) Dixmier trace, a central notion in Connes' non-commutative geometry [2]. Conversely, ideas in Connes' noncommutative geometry, such as measurability of operators [2, IV.2.…”

Section: Introductionmentioning

confidence: 99%

“…The paper is structured as follows: Section 1 introduces Banach limits, almost convergence (extending the notions of Lorentz [10]) and the theory of singular symmetric functionals on the Marcinkiewicz space M( ) defined by a concave function [4,5]. The construction of singular symmetric functionals on M( ) [5] (Definition 1.6 below) is extended by Definition 1.7.…”

Section: Introductionmentioning

confidence: 99%

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Dixmier traces as singular symmetric functionals and applications to measurable operators

Lord

Sedaev

Sukochev

2005

Journal of Functional Analysis

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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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Dixmier traces of operators on Banach and Hilbert spaces

Pietsch¹

2012

Mathematische Nachrichten

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We present a simple approach to the theory of traces on the Marcinkiewicz operator ideal Particular attention is paid to the Dixmier traces, which may be obtained from both dilation‐invariant functionals and shift‐invariant functionals on \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathfrak l_\infty$\end{document}. Since the concept of positivity does not make sense for traces defined on operator ideals over Banach spaces, traces are not assumed to be positive in the Hilbert space setting, as well. This standpoint raises some new problems. The present studies will be continued in 35, 36.

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Singular Symmetric Functionals

Cited by 23 publications

References 5 publications

Lidskii-Type Formulae for Dixmier Traces

Lidskii-Type Formulae for Dixmier Traces

Dixmier traces as singular symmetric functionals and applications to measurable operators

Dixmier traces of operators on Banach and Hilbert spaces

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