Singular Trace Research Articles

Endpoint weak Schatten class estimates and trace formula for commutators of Riesz transforms with multipliers on Heisenberg groups

Along the line of singular value estimates for commutators by Rochberg–Semmes, Lord–McDonald–Sukochev–Zanin and Fan–Lacey–Li, we establish the endpoint weak Schatten class estimate for commutators of Riesz transforms with multiplication operator Mf on Heisenberg groups via homogeneous Sobolev norm of the symbol f. The new tool we exploit is the construction of a singular trace formula on Heisenberg groups, which, together with the use of double operator integrals, allows us to bypass the use of Fourier analysis and provides a solid foundation to investigate the singular values estimates for similar commutators in general stratified Lie groups.

A New View at Dixmier Traces on $$\mathfrak {L}_{1,\infty } (H)$$ L 1 , ∞ ( H )

Dixmier traces are usually defined by means of functionals on $$\mathfrak {l}_\infty (\mathbb {N})$$ that are invariant under the forward dilation $$\begin{aligned} D_+: (\sigma _1,\sigma _2,\sigma _3,\sigma _4,\dots \;) \mapsto (\sigma _1,\sigma _1,\sigma _2,\sigma _2,\dots \;) \end{aligned}$$ and $$\mathfrak {c}_0$$ -singular, which means that they vanish at all null sequences. The backward dilation $$\begin{aligned} D_-: (\sigma _1,\sigma _2,\sigma _3,\sigma _4,\dots \;) \mapsto (\sigma _2,\sigma _4,\sigma _6,\sigma _8,\dots \;) \end{aligned}$$ can be used, as well. Note that $$D_-$$ -invariant functionals automatically vanish on $$\mathfrak {c}_0(\mathbb {N})$$ . As observed by Sukochev and coauthors, any requirement concerning dilation invariance can be dropped when working on $$\mathfrak {L}_{1,\infty } (H)$$ . It just suffices to assume that the generating functionals vanish on $$\mathfrak {c}_0(\mathbb {N})$$ . So it seems to be a good idea to present the theory of Dixmier traces on the bases of an adapted definition. We carry out this project. Although the fundamental results remain unchanged, their interplay becomes a quite different appearance. Even more is possible. Looking at the conclusions Hahn–Banach Theorem $$\mathfrak {c}_0$$ -singular functional Dixmier trace, we may wonder whether the intermediate step is necessary. Indeed, there exists a sublinear function on the underlying operator ideal $$\mathfrak {L}_{1,\infty } (H)$$ that directly yields all proper Dixmier traces via the Hahn–Banach Theorem. So the theory of Dixmier traces can be turned upside down:

Universal measurability and the Hochschild class of the Chern character

We study notions of measurability for singular traces, and characterise universal measurability for operators in Dixmier ideals. This measurability result is then applied to improve on the various proofs of Connes’ identication of the Hochschild class of the Chern character of Dixmier summable spectral triples. The measurability results show that the identication of the Hochschild class is independent of the choice of singular trace. As a corollary we obtain strong information on the asymptotics of the eigenvalues of operators naturally associated to spectral triples \mathcal A, H, D and Hochschild cycles for \mathcal A .

Traces on symmetrically normed operator ideals

For every symmetrically normed ideal $\mathcal{E}$ of compact operators, we give a criterion for the existence of a continuous singular trace on $\mathcal{E}$. We also give a criterion for the existence of a continuous singular trace on $\mathcal{E}$ which respects Hardy-Littlewood majorization. We prove that the class of all continuous singular traces on $\mathcal{E}$ is strictly wider than the class of continuous singular traces which respect Hardy-Littlewood majorization. We establish a canonical bijection between the set of all traces on $\mathcal{E}$ and the set of all symmetric functionals on the corresponding sequence ideal. Similar results are also proved in the setting of semifinite von Neumann algebras.

Traces of compact operators and the noncommutative residue

We extend the noncommutative residue of M. Wodzicki on compactly supported classical pseudo-differential operators of order −d and generalise A. Connes’ trace theorem, which states that the residue can be calculated using a singular trace on compact operators. Contrary to the role of the noncommutative residue for the classical pseudo-differential operators, a corollary is that the pseudo-differential operators of order −d do not have a ‘unique’ trace; pseudo-differential operators can be non-measurable in Connes’ sense. Other corollaries are given clarifying the role of Dixmier traces in noncommutative geometry, including the definitive statement of Connes’ original theorem.

Edge-boundary problems with singular trace conditions

The ellipticity of boundary value problems on a smooth manifold with boundary relies on a two-component principal symbolic structure \({(\sigma_{\psi},\sigma_\partial)}\), consisting of interior and boundary symbols. In the case of a smooth edge on manifolds with boundary, we have a third symbolic component, namely, the edge symbol \({\sigma_\wedge}\), referring to extra conditions on the edge, analogously as boundary conditions. Apart from such conditions ‘in integral form’ there may exist singular trace conditions, investigated in Kapanadze et al., Internal Equations and Operator Theory, 61, 241–279, 2008 on ‘closed’ manifolds with edge. Here, we concentrate on the phenomena in combination with boundary conditions and edge problem.

Operators with Singular Trace Conditions on a Manifold with Edges

We establish a new calculus of pseudodifferential operators on a manifold with smooth edges and study ellipticity with extra trace and potential conditions (as well as Green operators) at the edge. In contrast to the known scenario with conditions of that kind in integral form we admit in this paper ‘singular’ trace and Green operators. In contrast to standard conditions in the theory of elliptic boundary value problems (like Dirichlet or Neumann conditions) our singular trace conditions, in general, do not act on functions that are smooth up to the boundary, but admit a more general asymptotic structure. Their action is now associated with the Laurent coefficients of the meromorphic Mellin transforms of functions with respect to the half-axis variable, the distance to the edge.

A especial consideração do outro na virtude da justiça na ética do Aristóteles maduro

O texto examina um dos traços caracterizadores da justiça no catálogo aristotélico das virtudes éticas: a implicação do outro. A partir da constatação de que o outro é requerido em toda situação ética – e assim marca a constituição de toda virtude ética – com que sentido Aristóteles afirma ser a alteridade o traço distintivo da justiça em sentido específico?

Dimensions and singular traces for spectral triples, with applications to fractals

Given a spectral triple ( A, H,D) , the functionals on A of the form a↦ τ ω ( a| D| − α ) are studied, where τ ω is a singular trace, and ω is a generalised limit. When τ ω is the Dixmier trace, the unique exponent d giving rise possibly to a non-trivial functional is called Hausdorff dimension, and the corresponding functional the ( d-dimensional) Hausdorff functional. It is shown that the Hausdorff dimension d coincides with the abscissa of convergence of the zeta function of | D| −1, and that the set of α's for which there exists a singular trace τ ω giving rise to a non trivial functional is an interval containing d. Moreover, the endpoints of such traceability interval have a dimensional interpretation. The functionals corresponding to points in the traceability interval are called Hausdorff–Besicovitch functionals. These definitions are tested on fractals in R , by computing the mentioned quantities and showing in many cases their correspondence with classical objects. In particular, for self-similar fractals the traceability interval consists only of the Hausdorff dimension, and the corresponding Hausdorff–Besicovitch functional gives rise to the Hausdorff measure. More generally, for any limit fractal, the described functionals do not depend on the generalized limit ω.

ON THE DOMAIN OF SINGULAR TRACES

The question whether an operator belongs to the domain of some singular trace is addressed, together with the dual question whether an operator does not belong to the domain of some singular trace. We show that the answers are positive in general, namely for any (compact, infinite rank) positive operator A we exhibit two singular traces, the first being zero and the second being infinite on A. However, if we assume that the singular traces are genrated by a "regular" operator, the answers change, namely such traces alway vanish on trace-class, non singularly traceable operators and are always infinite on non trace-class, non singularly traceable operators. These results are achieved on a general semifinite factor and make use of a new characterization of singular traceability (cf. [7]).

Singular Traces and Compact Operators

We give a necessary and sufficient condition on a positive compact operatorTfor the existence of a singular trace (i.e. a trace vanishing on the finite rank operators) which takes a finite non-zero value onT. This generalizes previous results by Dixmier and Varga. We also give an explicit description of these traces and associated ergodic states onl∞(N) using tools of non standard analysis in an essential way.

Singular Traces on Semifinite von Neumann Algebras

Singular traces are constructed on a general semifinite von Neumann algebra, thus generalizing the result of Dixmier (C.R. Acad. Sci. Paris262, 1966.) Moreover our technique produces singular traces on type II1 factors. Such traces, though vanishing on all bounded operators, are non trivial on the *-algebra of affiliated unbounded operators. On a semifinite factor, we show that all traces are given by a dilation invariant functional on the cone of positive decreasing functions on [0, ∞), and we prove that the existence of a singular trace which is non trivial on a given operator is equivalent to an eccentricity condition on the singular values function, a result which generalizes the theorem given in (S. Albeverio, D. Guido, A. Ponosov, and S. Scarlatti, J. Funct. Anal., to appear.) for B(H).

On the semi-hyponormal n-tuple of operators

The theory of singular integral model and trace formula is extended to the context of hyponormal or semi-hyponormal n-tuple of operators. The spectrum of noncommutative n-tuple of operators is examined.