Lidskii-Type Formulae for Dixmier Traces

Sedaev, A. A.; Sukochev, Fedor; Zanin, Dmitriy

doi:10.1007/s00020-010-1828-1

Cited by 13 publications

(20 citation statements)

References 17 publications

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“…The following theorem is a Lidskii formula for all traces on the ideal L 1,∞ . This result extends and complements the corresponding results from [1,5,6,50,55].…”

Section: Lidskii Formulasupporting

confidence: 88%

Banach limits and traces on L1,∞

Семенов

Sukochev

Usachev

et al. 2015

Advances in Mathematics

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We introduce a new approach to traces on the principal ideal L 1,∞ generated by any positive compact operator whose singular value sequence is the harmonic sequence. Distinct from the well-known construction of J. Dixmier, the new approach provides the explicit construction of every trace of every operator in L 1,∞ in terms of translation invariant functionals applied to a sequence of restricted sums of eigenvalues. The approach is based on a remarkable bijection between the set of all traces on L 1,∞ and the set of all translation invariant functionals on l ∞ . This bijection allows us to identify all known and commonly used subsets of traces (Dixmier traces, Connes-Dixmier traces, etc.) in terms of invariance properties of linear functionals on l ∞ , and definitively classify the measurability of operators in L 1,∞ in terms of qualified convergence of sums of eigenvalues. This classification has led us to a resolution of several open problems (for the class L 1,∞ ) from [7]. As an application we extend Connes' classical trace theorem to positive normalised traces.

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“…The following theorem is a Lidskii formula for all traces on the ideal L 1,∞ . This result extends and complements the corresponding results from [1,5,6,50,55].…”

Section: Lidskii Formulasupporting

confidence: 88%

Banach limits and traces on L1,∞

Семенов

Sukochev

Usachev

et al. 2015

Advances in Mathematics

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“…It was also shown in [17,Theorem 5] that there exists a Dixmier trace Tr ω such that the formula (3) does not hold.…”

Section: Introductionmentioning

confidence: 99%

Dixmier traces generated by exponentiation invariant generalised limits

Sukochev¹,

Usachev²,

Zanin³

2014

J. Noncommut. Geom.

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“…residue and heat kernel results are dependent on submajorisation (cf. proofs in [15] and [33]). It is open whether this dependence is a necessity for the results or an artifact of the proofs.…”

Section: Closed Symmetric Subidealsmentioning

confidence: 96%

Measure Theory in Noncommutative Spaces

Lord

Sukochev

2010

SIGMA

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The integral in noncommutative geometry (NCG) involves a non-standard trace called a Dixmier trace. The geometric origins of this integral are well known. From a measure-theoretic view, however, the formulation contains several difficulties. We review results concerning the technical features of the integral in NCG and some outstanding problems in this area. The review is aimed for the general user of NCG

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Lidskii-Type Formulae for Dixmier Traces

Abstract: We establish several analogues of the classical Lidskii Theorem for some special classes of singular traces (Dixmier traces and ConnesDixmier traces) used in noncommutative geometry. Mathematics Subject Classification (2010). 46L52, 47B10, 46E30.

Cited by 13 publications

References 17 publications

Banach limits and traces on L1,∞

Banach limits and traces on L1,∞

Dixmier traces generated by exponentiation invariant generalised limits

Measure Theory in Noncommutative Spaces

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