Dixmier traces and extrapolation description of noncommutative Lorentz spaces

Gayral, Victor; Sukochev, Fedor

doi:10.1016/j.jfa.2014.02.036

Cited by 8 publications

(17 citation statements)

References 62 publications

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“…We conclude that for a suitable C > 0, β t ≥ Ct 1 2 +ǫ . Moreover, the estimate (16) implies that for large t we have…”

Section: Symbol Properties Of Modulated Operatorsmentioning

confidence: 99%

“…Remark 8. 16. The reader should note that we are using a diffeomorphism φ : Diag 2ǫ → T M since we do not wish to concern ourselves with the boundary behaviour at the boundary of the tubular neighborhood.…”

Section: Localizing ϕ-Modulated Operators On Manifoldsmentioning

confidence: 99%

“…The assertion follows from Theorem 9.1. It is proved in [16,Theorem 3.3] that for ϕ(t) = log k (e+t) e+t , every positive G ∈ L ϕ and every exponentiation invariant extended limit ω the following formula holds:…”

Section: Connes' Trace Formula On Closed Manifoldsmentioning

confidence: 99%

“…The residue at this pole computes the higher Wodzicki residue. Another way to look at the ζfunction is to consider the expression Tr(A 1+s ) for a positive operator A [6,37,16]. We show that for log-classical pseudo-differential operators the higher Wodzicki residue can be computed using this function, too.…”

Section: Introductionmentioning

confidence: 97%

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Dixmier traces and residues on weak operator ideals

Goffeng

Usachev

2020

Journal of Mathematical Analysis and Applications

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We develop the theory of modulated operators in general principal ideals of compact operators. For Laplacian modulated operators we establish Connes' trace formula in its local Euclidean model and a global version thereof. It expresses Dixmier traces in terms of the vector-valued Wodzicki residue. We demonstrate the applicability of our main results in the context of log-classical pseudo-differential operators, studied by Lesch, and a class of operators naturally appearing in noncommutative geometry.where Res W (A) is the Wodzicki residue of A and Tr ω is any Dixmier trace.This theorem reduces the computation of Dixmier traces to that of the Wodzicki residue, which in turn is defined from the principal symbol of an operator and is therefore more accessible to computations. The spectral asymptotics of positive pseudo-differential operators was well known prior to Connes' trace formula by means of a Weyl law, see for instance [22, Chapter 29.1], but Connes' trace formula conceptualized Dixmier traces as an "integral" in noncommutative geometry. There were several attempts to extend and generalise Connes' trace formula (see e.g. [3], [15] and [19] for the treatment of anisotropic pseudo-differential operators, the perturbed Laplacian on the noncommutative two tori and manifolds with boundaries, respectively). Recent developments.A recent extension of the above theorem appeared in [23] (see also [26]). It expresses the Dixmier traces of so-called Laplacian modulated operators (for rigorous definitions, see Section 4 below) in terms of a vector-valued Wodzicki residue, which again depends on the symbol of an operator. We state the main result of [23] here. Let ℓ ∞ (N) denote the Banach space of bounded sequences with supremum norm. Example 2.8. Let ϕ be a function satisfying (3) and (M, g) a closed Riemannian manifold. The Weyl law guarantees the eigenvalue asymptotics λ k ((1+∆ g ) d/2 ) = c g k+o(k) for a suitable constant c g > 0. Condition (3) implies that ϕ((1 − ∆ g ) d/2 ) ∈ L ϕ . Example 2.9. If (M, g) is a Riemannian manifold of dimension d, and ρ g denotes the geodesic distance, we can define an operator T : C ∞ c (M) → C ∞ (M) by (7) T f (x) := M f (x) − f (y) ρ g (x, y) d dV g ,

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“…We conclude that for a suitable C > 0, β t ≥ Ct 1 2 +ǫ . Moreover, the estimate (16) implies that for large t we have…”

Section: Symbol Properties Of Modulated Operatorsmentioning

confidence: 99%

Section: Localizing ϕ-Modulated Operators On Manifoldsmentioning

confidence: 99%

Section: Connes' Trace Formula On Closed Manifoldsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

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Dixmier traces and residues on weak operator ideals

Goffeng

Usachev

2020

Journal of Mathematical Analysis and Applications

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“…For the case of L ψ (T) one can refer to [60,Lemma 15] to remove this assumption. Below we will need the relation between generalised heat kernels and Dixmier traces on L ψ (T), which was proved in [25] under the additional assumption that…”

Section: The Kms-state φ ω and Dixmier Tracesmentioning

confidence: 99%

Constructing KMS states from infinite-dimensional spectral triples

Goffeng

Rennie

Usachev

2019

Journal of Geometry and Physics

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We construct KMS-states from Li 1 -summable semifinite spectral triples and show that in several important examples the construction coincides with well-known direct constructions of KMS-states for naturally defined flows. Under further summability assumptions the constructed KMS-state can be computed in terms of Dixmier traces. For closed manifolds, we recover the ordinary Lebesgue integral. For Cuntz-Pimsner algebras with their gauge flow, the construction produces KMS-states from traces on the coefficient algebra and recovers the Laca-Neshveyev correspondence. For a discrete group acting on its Stone-Cech boundary, we recover the Patterson-Sullivan measures on the Stone-Čech boundary for a flow defined from the Radon-Nikodym cocycle. *

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