arXiv subjects

Raphael Ponge

Publications and source records attributed to Raphael Ponge.

At least 19 recordsLinked to original sources

arXiv2026

Nonclassical Weyl laws and Connes' Integration for weak Lorentz ideals, II

This is the second in a series of papers on Connes' integration in weak Lorentz ideals. Building on the Dixmier trace theory, Birman--Solomyak perturbation theory, and strong measurability developed in Part 1, we establish a spectral form of Pietsch's correspondence for traces on these ideals, extending to this setting results of Semenov--Sukochev--Usachev--Zanin for the weak trace-class. The correspondence describes the positive normalized traces in terms of Banach limits and yields a complete spectral characterization of strong measurability. We also introduce hypermeasurability (measurability with respect to every normalized trace). Unlike the ambient ideal, it depends on the specific regularly varying function chosen, and we characterize it spectrally by means of eigenvalue sums. We further show that hypermeasurability and spectral measurability are incomparable. Finally, we apply these results to examples arising from nonclassical Weyl laws in the sense of Simon, including the logarithm of the Laplacian on a closed manifold, the double Laplacian, multi-tensor products of Laplacians, and an example arising from Connes' approach to the Riemann Hypothesis.

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arXiv2026

Functional Calculus on Noncommutative Tori, II. Complex Powers, Logarithms, and Sectorial Projections

This paper develops a systematic theory of complex powers, logarithms, and sectorial projections of elliptic pseudodifferential operators on noncommutative tori, extending to this setting the classical constructions of Seeley and others. Building on the parametric pseudodifferential calculus of the prequel~\cite{LP:Part1}, we construct the complex powers associated with a given ray, show that they form a holomorphic family of pseudodifferential operators with the semigroup property, and compute their symbols. We further establish exponential growth bounds on vertical strips in the operator, Schatten, and trace-class topologies by means of a new holomorphic calculus for pseudodifferential families. The logarithm is identified as a pseudodifferential operator whose symbol is determined by the resolvent symbol, and the associated trace formula is derived. Sectorial projections are constructed as contour integrals and shown to be of order zero. This yields analogues for noncommutative tori of results of Wodzicki, Okikiolu, and Gaarde--Grubb, and provides the analytic foundations for the spectral-geometric applications developed in subsequent papers.

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arXiv2026

Nonclassical Weyl laws and Connes' Integration for weak Lorentz ideals, I

Motivated by nonclassical Weyl laws arising in various contexts (including Connes' approach to the Riemann Hypothesis), we develop a systematic theory of Dixmier traces and Connes' noncommutative integration for weak Lorentz ideals associated with regularly varying functions. A key ingredient is an asymptotic additivity property for eigenvalue partial sums, obtained by combining Karamata's theorem with results of Kalton and Lord-Sukochev-Zanin. This yields a direct construction of Dixmier traces in terms of eigenvalue sequences and a complete spectral characterization of measurable operators, answering a question of Connes in this general setting. We also extend to weak Lorentz ideals the Birman-Solomyak perturbation theory for eigenvalue and singular-value asymptotics. Weyl operators (those admitting precise asymptotic limits for their rescaled eigenvalue sequences) are shown to form a closed subset of the ideal, stable under compact perturbations, extending classical results of Weyl and Birman-Solomyak. We further study strong measurability (measurability with respect to all positive normalized traces). We prove that every Weyl operator is strongly measurable, so spectral measurability implies strong measurability. The converse does not hold in general; a spectral characterization via Pietsch's correspondence is obtained in the forthcoming companion pape. Finally, as an application, we establish spectral measurability for operators arising from nonclassical Weyl laws: operators associated with the Riemann Hypothesis (assuming RH); Schr\"odinger operators with anisotropic potentials and Dirichlet Laplacians on infinite-volume domains; Dirac operators on open spin manifolds with conformally cusp metrics; and the operator formed by the Dirac operator of the Podle\`s quantum sphere and the Laplacian on the 2-torus.

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arXiv2026

Noncommutative Geometry, Spectral Asymptotics, and Semiclassical Analysis

Semiclassical analysis and noncommutative geometry are two pillars of quantum theory. It is only recently that bridges between them have been emerging. In this monograph, we combine various techniques from functional analysis and spectral theory to obtain semiclassical Weyl laws and extensions of Connes' integration formula for a large class of noncommutative manifolds (i.e., spectral triples). These results generalize and simplify recent results of McDonald-Sukochev-Zanin. In particular, all the regularity assumptions and restrictions on dimension there are removed in our approach. Moreover, the Tauberian condition used by McDonald-Sukochev-Zanin is replaced by a weaker spectral theoretic condition, called Condition (W). That condition holds in fairly greater generality and significantly opens the scope of applicability of the main results. We also give Tauberian conditions that imply Condition (W). These Tauberian conditions are easier to check in practice than the Tauberian condition of McDonald-Sukochev-Zanin and are satisfied in numerous examples. The need for these conditions was highlighted by Alain Connes in an online seminar. The main results of this paper are illustrated by semiclassical Weyl laws and integration formulas in the settings of closed Riemannian manifolds and quantum tori. In the former settings we recover well-known semiclassical Weyl laws, as well as Weyl laws for Steklov eigenvalues. The only novelty is obtaining them from old results of Minakshisundaram-Pleijel on heat kernel asymptotics. In the setting of quantum tori, the semiclassical Weyl laws provide a positive answer to a conjecture of Edward McDonald and the author. The integration formulas are refinements of several previous analogues of Connes' integration formula for quantum tori.

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arXiv2021

Weyl's laws and Connes' integration formulas for matrix-valued $L\log L$-Orlicz potentials

Thanks to the Birman-Schwinger principle, Weyl's laws for Birman-Schwinger operators yields semiclassical Weyl's laws for the corresponding Schr\"odinger operators. In a recent preprint Rozenblum established quite general Weyl's laws for Birman-Schwinger operators associated with pseudodifferential operators of critical order and potentials that are product of $L\log L$-Orlicz functions and Alfhors-regular measures supported on a submanifold. In this paper, for matrix-valued $L\log L$-Orlicz potentials supported on the whole manifold, Rozenblum's results are direct consequences of the Cwikel-type estimates on tori recently established by Sukochev-Zanin. As applications we obtain CLR-type inequalities and semiclassical Weyl's laws for critical Schr\"odinger operators associated with matrix-valued$L\log L$-Orlicz potentials. Finally, we explain how the Weyl's laws of this paper imply a strong version of Connes' integration formula for matrix-valued $L\log L$-Orlicz potentials.

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arXiv2021

Connes' integration and Weyl's laws

This paper deal with some questions regarding the notion of integral in the framework of Connes's noncommutative geometry. First, we present a purely spectral theoretic construction of Connes' integral. This answers a question of Alain Connes. We also deal with the compatibility of Dixmier traces with Lebesgue's integral. This answers another question of Alain Connes. We further clarify the relationship of Connes' integration with Weyl's laws for compact operators and Birman-Solomyak's perturbation theory. We also give a "soft proof" of Birman-Solomyak's Weyl's law for negative order pseudodifferential operators on closed manifold. This Weyl's law yields a stronger form of Connes' trace theorem. Finally, we explain the relationship between Connes' integral and semiclassical Weyl's law for Schroedinger operators. This is an easy consequence of the Birman-Schwinger principle. We thus get a neat link between noncommutative geometry and semiclassical analysis.

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arXiv2021

Dixmier Trace Formulas and Negative Eigenvalues of Schroedinger Operators on Curved Noncommutative Tori

In a previous paper we established Cwikel-type estimates on noncommutative tori and used them to get analogues in this setting of the Cwikel-Lieb-Rozenblum (CLR) and Lieb-Thirring inequalities for negative eigenvalues of fractional Schr\"odinger operators. In this paper, we focus on "curved" NC tori, where the role of the usual Laplacian is played by Laplace-Beltrami operators associated with arbitrary Riemannian metrics. The Cwikel-type estimates of our previous paper are extended to pseudodifferential operators and powers of Laplace-Beltrami operators. There are several applications of these estimates. First, we get $L_p$-versions of the usual formula for the trace of \psidos\ on NC tori, i.e., for combinations of \psidos\ with $L_p$-position operators. Next, we get $L_p$-versions of the analogues for NC tori Connes' trace theorem and Connes' integration formula. They give formulas for the NC integrals (a.k.a.\ Dixmier traces) of products of $L_p$-position operators with \psidos\ or powers of the Laplace-Beltrami operators. Moreover, by combining our Cwikel-type estimates with suitable versions of the Birman-Schwinger principle we get versions of the CLR and Lieb-Thirring inequalities for negative eigenvalues of fractional Schr\"odinger operators associated with powers of Laplace-Beltrami operators and $L_p$-potentials. As in the original Euclidean case the Lieb-Thirring inequalities imply a dual Sobolev inequality for orthonormal families. Finally, we discuss spectral asymptotics and semiclassical Weyl's laws for the our classes of operators on curved NC tori. This superseded a previous conjecture in our previous paper.

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arXiv2021

Cwikel Estimates and Negative Eigenvalues of Schroedinger Operators on Noncommutative Tori

In this paper, we establish Cwikel-type estimates for noncommutative tori for any dimension~$n\geq 2$. We use them to derive Cwikel-Lieb-Rozenblum inequalities and and Lieb-Thirring inequalities for the number of negative eigenvalues of fractional Schroedinger operators on noncommutative tori in any dimension~$n\geq 2$. The latter leads to a Sobolev inequality for noncommutative tori. On the way we establish a "borderline version" of the abstract Birman-Schwinger principle for the number of negative eigenvalues of relatively compact form perturbations of a non-negative semi-bounded operator with isolated 0-eigenvalue.

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arXiv2019

Connes Trace Theorem for Curved Noncommutative Tori. Application to Scalar Curvature

In this paper we prove a version of Connes' trace theorem for noncommutative tori of any dimension~$n\geq 2$. This allows us to recover and improve earlier versions of this result in dimension $n=2$ and $n=4$ by Fathizadeh-Khalkhali. We also recover the Connes integration formula for flat noncommutative tori of McDonald-Sukochev-Zanin. As a further application we prove a curved version of this integration formula in terms of the Laplace-Beltrami operator defined by an arbitrary Riemannian metric. For the class of so-called self-compatible Riemannian metrics (including the conformally flat metrics of Connes-Tretkoff) this shows that Connes' noncommutative integral allows us to recover the Riemannian density. This exhibits a neat link between this notion of noncommutative integral and noncommutative measure theory in the sense of operator algebras. As an application of these results, we setup a natural notion of scalar curvature for curved noncommutative tori.

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arXiv2019

Functional Calculus for Elliptic Operators on Noncommutative Tori, I

In this paper, we introduce a parametric pseudodifferential calculus on noncommutative $n$-tori which is a natural nest for resolvents of elliptic pseudodifferential operators. Unlike in some previous approaches to parametric pseudodifferential calculi, our parametric pseudodifferential calculus contains resolvents of elliptic pseudodifferential operators that need not be differential operators. As an application we show that complex powers of positive elliptic pseudodifferential operators on noncommutative $n$-tori are pseudodifferential operators. This confirms a claim of Fathi-Ghorbanpour-Khalkhali.

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arXiv2019

Laplace-Beltrami Operators on Noncommutative Tori

In this paper, we construct Laplace-Beltrami operators associated with arbitrary Riemannian metrics on noncommutative tori of any dimension. These operators enjoy the main properties of the Laplace-Beltrami operators on ordinary Riemannian manifolds. The construction takes into account the non-triviality of the group of modular automorphisms. On the way we introduce notions of Riemannian density and Riemannian volumes for noncommutative tori.

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arXiv2018

Periodicity and Cyclic Homology. Para-S-Modules and Perturbation Lemmas

In this paper, we introduce a paracyclic version of $S$-modules. These new objects are called para-$S$-modules. Paracyclic modules and parachain complexes give rise to para-$S$-modules much in the same way as cyclic modules and mixed complexes give rise to $S$-modules. More generally, para-$S$-modules provide us with a natural framework to get analogues for paracyclic modules and parachain complexes of various constructions and equivalence results for cyclic modules or mixed complexes. The datum of a para-$S$-module does not provide us with a chain complex, and so notions of homology and quasi-isomorphisms do not make sense. We establish some generalizations for para-$S$-modules and parachain complexes of the basic perturbation lemma of differential homological algebra. These generalizations provide us with general recipes for converting deformation retracts of Hoschschild chain complexes into deformation retracts of para-$S$-modules. By using ideas of Kassel this then allows us to get comparison results between the various para-$S$-modules associated with para-precyclic modules, and between them and Connes' cyclic chain complex. These comparison results lead us to alternative descriptions of Connes' periodicity operator. This has some applications in periodic cyclic homology. We also describe the counterparts of these results in cyclic cohomology. In particular, we obtain an explicit way to convert a periodic $(b,B)$-cocycle into a cohomologous periodic cyclic cocycle.

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arXiv2018

Pseudodifferential calculus on noncommutative tori, I. Oscillating integrals

This paper is the first part of a two-paper series whose aim is to give a thorough account on Connes' pseudodifferential calculus on noncommutative tori. This pseudodifferential calculus has been used in numerous recent papers, but a detailed description is still missing. In this paper, we focus on constructing an oscillating integral for noncommutative tori and laying down the main functional analysis ground for understanding Connes' pseudodifferential calculus. In particular, this allows us to give a precise explanation of the definition of pseudodifferential operators on noncommutative tori. More generally, this paper introduces the main technical tools that are used in the 2nd part of the series to derive the main properties of these operators. In addition, we establish the equivalence between our class of operators and the toroidal pseudo differential operators considered by other authors.

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arXiv2018

Pseudodifferential calculus on noncommutative tori, II. Main properties

This paper is the 2nd part of a two-paper series whose aim is to give a detailed description of Connes' pseudodifferential calculus on noncommutative $n$-tori, $n\geq 2$. We make use of the tools introduced in the 1st part to deal with the main properties of pseudodifferential operators on noncommutative tori of any dimension $n\geq 2$. This includes the main results mentioned in the original notes of Connes and Baaj. We also obtain further results regarding action on Sobolev spaces, spectral theory of elliptic operators, and Schatten-class properties of pseudodifferential operators of negative order, including a trace-formula for pseudodifferential operators of order $<-n$.

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arXiv2017

Privileged Coordinates and Nilpotent Approximation of Carnot Manifolds, I. General Results

In this paper we attempt to give a systematic account on privileged coordinates and the nilpotent approximation of Carnot manifolds. By a Carnot manifold it is meant a manifold with a distinguished filtration of subbundles of the tangent bundle which is compatible with the Lie bracket of vector fields. This paper lies down the background for its sequel by clarifying a few points on privileged coordinates and the nilpotent approximation of Carnot manifolds. In particular, we give a description of all the systems of privileged coordinates at a given point. We also give an algebraic characterization of all nilpotent groups that appear as the nilpotent approximation at a given point. In fact, given a nilpotent group $G$ satisfying this algebraic characterization, we exhibit all the changes of variables that transform a given system of privileged coordinates into another system of privileged coordinates in which the nilpotent approximation is given by $G$.

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arXiv2017

Cyclic Homology and Group Actions

In this paper we present the construction of explicit quasi-isomorphisms that compute the cyclic homology and periodic cyclic homology of crossed-product algebras associated with (discrete) group actions. In the first part we deal with algebraic crossed-products associated with group actions on unital algebras over any ring $k\supset \mathbb{Q}$. In the second part, we extend the results to actions on locally convex algebras. We then deal with crossed-products associated with group actions on manifolds and smooth varieties. For the finite order components, the results are expressed in terms of what we call "mixed equivariant cohomology". This "mixed" theory mediates between group homology and de Rham cohomology. It is naturally related to equivariant cohomology, and so we obtain explicit constructions of cyclic cycles out of equivariant characteristic classes. For the infinite order components, we simplify and correct the misidentification of Crainic. An important new homological tool is the notion of "triangular $S$-module". This is a natural generalization of the cylindrical complexes of Getzler-Jones. It combines the mixed complexes of Burghelea-Kassel and parachain complexes of Getzler-Jones with the $S$-modules of Kassel-Jones. There are spectral sequences naturally associated with triangular $S$-modules. In particular, this allows us to recover spectral sequences Feigin-Tsygan and Getzler-Jones and leads us to a new spectral sequence.

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arXiv2017

Privileged Coordinates and Nilpotent Approximation for Carnot Manifolds, II. Carnot Coordinates

This paper is a sequel of arxiv:1709.09045 and deals with privileged coordinates and nilpotent approximation of Carnot manifolds. By a Carnot manifold it is meant a manifold equipped with a filtration by subbundles of the tangent bundle which is compatible with the Lie bracket of vector fields. In this paper, we single out a special class of privileged coordinates in which the nilpotent approximation at a given point of a Carnot manifold is given by its tangent group. We call these coordinates Carnot coordinates. Examples of Carnot coordinates include Darboux coordinates on contact manifolds and the canonical coordinates of the first kind of Goodman and Rothschild-Stein. By converting the privileged coordinate of Bella\"iche into Carnot coordinates we obtain an effective construction of Carnot coordinates, which we call $\varepsilon$-Carnot coordinates. They form the building block of all systems of Carnot coordinates. On a graded nilpotent Lie group they are given by the group law of the group. For general Carnot manifolds, they depend smoothly on the base point. Moreover, in Carnot coordinates at a given point, they are osculated in a very precise manner by the group law of the tangent group at the point.

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arXiv2015

Tangent Maps and Tangent Groupoid for Carnot Manifolds

This paper studies the infinitesimal structure of Carnot manifolds. By a Carnot manifold we mean a manifold together with a subbundle filtration of its tangent bundle which is compatible with the Lie bracket of vector fields. We introduce a notion of differential, called Carnot differential, for Carnot manifolds maps (i.e., maps that are compatible with the Carnot manifold structure). This differential is obtained as a group map between the corresponding tangent groups. We prove that, at every point, a Carnot manifold map is osculated in a very precise way by its Carnot differential at the point. We also show that, in the case of maps between nilpotent graded groups, the Carnot differential is given by the Pansu derivative. Therefore, the Carnot differential is the natural generalization of the Pansu derivative to maps between general Carnot manifolds. Another main result is a construction of an analogue for Carnot manifolds of Connes' tangent groupoid. Given any Carnot manifold $(M,H)$ we get a smooth groupoid that encodes the smooth deformation of the pair $M\times M$ to the tangent group bundle $GM$. This shows that, at every point, the tangent group is the tangent space in a true differential-geometric fashion. Moreover, the very fact that we have a groupoid accounts for the group structure of the tangent group. Incidentally, this answers a well-known question of Bella\"iche.

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