The Experts below are selected from a list of 234 Experts worldwide ranked by ideXlab platform
Masoud Khalkhali - One of the best experts on this subject based on the ideXlab platform.
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Weyl’s Law and Connes’ Trace Theorem for Noncommutative Two Tori
Letters in Mathematical Physics, 2013Co-Authors: Farzad Fathizadeh, Masoud KhalkhaliAbstract:We prove the analogue of Weyl’s law for a noncommutative Riemannian manifold, namely the noncommutative two torus $${\mathbb{T}_{\theta}^{2}}$$ equipped with a general translation invariant conformal structure and a Weyl conformal factor. This is achieved by studying the asymptotic distribution of the eigenvalues of the perturbed Laplacian on $${\mathbb{T}_{\theta}^{2}}$$ . We also prove the analogue of Connes’ Trace Theorem by showing that the Dixmier Trace and a noncommutative residue coincide on pseudodifferential operators of order −2 on $${\mathbb{T}_{\theta}^{2}}$$ .
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weyl s law and connes Trace Theorem for noncommutative two tori
Letters in Mathematical Physics, 2013Co-Authors: Farzad Fathizadeh, Masoud KhalkhaliAbstract:We prove the analogue of Weyl’s law for a noncommutative Riemannian manifold, namely the noncommutative two torus \({\mathbb{T}_{\theta}^{2}}\) equipped with a general translation invariant conformal structure and a Weyl conformal factor. This is achieved by studying the asymptotic distribution of the eigenvalues of the perturbed Laplacian on \({\mathbb{T}_{\theta}^{2}}\) . We also prove the analogue of Connes’ Trace Theorem by showing that the Dixmier Trace and a noncommutative residue coincide on pseudodifferential operators of order −2 on \({\mathbb{T}_{\theta}^{2}}\) .
Farzad Fathizadeh - One of the best experts on this subject based on the ideXlab platform.
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Weyl’s Law and Connes’ Trace Theorem for Noncommutative Two Tori
Letters in Mathematical Physics, 2013Co-Authors: Farzad Fathizadeh, Masoud KhalkhaliAbstract:We prove the analogue of Weyl’s law for a noncommutative Riemannian manifold, namely the noncommutative two torus $${\mathbb{T}_{\theta}^{2}}$$ equipped with a general translation invariant conformal structure and a Weyl conformal factor. This is achieved by studying the asymptotic distribution of the eigenvalues of the perturbed Laplacian on $${\mathbb{T}_{\theta}^{2}}$$ . We also prove the analogue of Connes’ Trace Theorem by showing that the Dixmier Trace and a noncommutative residue coincide on pseudodifferential operators of order −2 on $${\mathbb{T}_{\theta}^{2}}$$ .
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weyl s law and connes Trace Theorem for noncommutative two tori
Letters in Mathematical Physics, 2013Co-Authors: Farzad Fathizadeh, Masoud KhalkhaliAbstract:We prove the analogue of Weyl’s law for a noncommutative Riemannian manifold, namely the noncommutative two torus \({\mathbb{T}_{\theta}^{2}}\) equipped with a general translation invariant conformal structure and a Weyl conformal factor. This is achieved by studying the asymptotic distribution of the eigenvalues of the perturbed Laplacian on \({\mathbb{T}_{\theta}^{2}}\) . We also prove the analogue of Connes’ Trace Theorem by showing that the Dixmier Trace and a noncommutative residue coincide on pseudodifferential operators of order −2 on \({\mathbb{T}_{\theta}^{2}}\) .
Raphael Ponge - One of the best experts on this subject based on the ideXlab platform.
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connes s Trace Theorem for curved noncommutative tori application to scalar curvature
Journal of Mathematical Physics, 2020Co-Authors: Raphael PongeAbstract:In this paper, we prove a version of Connes’s Trace Theorem for noncommutative tori of any dimension n ⩾ 2. This allows us to recover and improve earlier versions of this result in dimensions n = 2...
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Connes’s Trace Theorem for curved noncommutative tori: Application to scalar curvature
Journal of Mathematical Physics, 2020Co-Authors: Raphael PongeAbstract:In this paper, we prove a version of Connes’s Trace Theorem for noncommutative tori of any dimension n ⩾ 2. This allows us to recover and improve earlier versions of this result in dimensions n = 2...
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connes Trace Theorem for curved noncommutative tori application to scalar curvature
arXiv: Operator Algebras, 2019Co-Authors: Raphael PongeAbstract: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.
Dmitriy Zanin - One of the best experts on this subject based on the ideXlab platform.
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conformal Trace Theorem for julia sets of quadratic polynomials
Ergodic Theory and Dynamical Systems, 2019Co-Authors: Alain Connes, Fedor Sukochev, Edward Mcdonald, Dmitriy ZaninAbstract:If $c$ is in the main cardioid of the Mandelbrot set, then the Julia set $J$ of the map $\unicode[STIX]{x1D719}_{c}:z\mapsto z^{2}+c$ is a Jordan curve of Hausdorff dimension $p\in [1,2)$ . We provide a full proof of a formula for the Hausdorff measure on $J$ in terms of singular Traces announced by the first named author in 1996.
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A $C^*$-algebraic approach to the principal symbol II.
arXiv: Operator Algebras, 2018Co-Authors: Fedor Sukochev, Edward Mcdonald, Dmitriy ZaninAbstract:We introduce an abstract theory of the principal symbol mapping for pseudodifferential operators extending the results of a preceding paper and providing a simple algebraic approach to the theory of pseudodifferential operators in settings important in noncommutative geometry. We provide a variant of Connes' Trace Theorem which applies to certain noncommutative settings, with a minimum of technical preliminaries. Our approach allows us to consider in a operators with non-smooth symbols, and we demonstrate the power of our approach by extending Connes' Trace Theorem to operators with non-smooth symbols in three examples: the Lie group $\mathrm{SU}(2)$, noncommutative tori and Moyal planes.
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Trace Theorem for quasi fuchsian groups
Sbornik Mathematics, 2017Co-Authors: Alain Connes, Fedor Sukochev, Dmitriy ZaninAbstract:We complete the proof of the Trace Theorem in the quantized calculus for quasi-Fuchsian group which was stated and sketched, but not fully proved, on pp. 322-325 in the book "Noncommutative Geometry" of the first author.
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krein s Trace Theorem revisited
arXiv: Operator Algebras, 2016Co-Authors: Denis Potapov, Fedor Sukochev, Dmitriy ZaninAbstract:We supply the first proof of Krein's Trace Theorem which does not use complex analysis. Our proof holds for~$\sigma$-finite von Neumann algebras $\mathcal{M}$ of type II and unbounded perturbations from the predual of~$\mathcal{M}$.
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krein s Trace Theorem revisited
Journal of Spectral Theory, 2014Co-Authors: Denis Potapov, Fedor Sukochev, Dmitriy ZaninAbstract:M. G. Krein’s celebrated Trace Theorem” states that if A ,B are self-adjoint operators in a separable Hilbert space such that A − B is a Trace class operator, then, for any function f of a real variable, whose derivative f ′ in distributional sense has Fourier transform belonging to L1(R) , the difference f(A)− f(B) is again a Trace class operator and the formula Trace ( f(A)− f(B) ) = ∫ R f ′(s)ξ(s)ds holds where the function ξ ∈ L1(R) depends only on A and B and is uniquely determined by the above formula. The function ξ is called the spectral shift function of the pair A ,B and is an important ingredient in the perturbation theory of self-adjoint operators. The original proof of Krein uses complex analysis and is quite involved. We supply a new proof which does not use complex analysis. Our proof works also for σ-finite von Neumann algebrasM of type II and unbounded perturbations from the predual ofM. The exposition is based on joint work with D. Potapov and D. Zanin. E-mail address: f.sukochev@unsw.edu.au School of Mathematics & Statistics, University of NSW, Kensington NSW 2052 AUSTRALIA
Mitsuru Sugimoto - One of the best experts on this subject based on the ideXlab platform.
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Stability of Trace Theorems on the sphere
arXiv: Classical Analysis and ODEs, 2016Co-Authors: Chris Jeavons, Tohru Ozawa, Mitsuru SugimotoAbstract:We prove stable versions of Trace Theorems on the sphere in $L^2$ with optimal constants, thus obtaining rather precise information regarding near-extremisers. We also obtain stability for the Trace Theorem into $L^q$ for $q > 2$, by combining a refined Hardy-Littlewood-Sobolev inequality on the sphere with a duality-stability result proved very recently by Carlen. Finally, we extend a local version of Carlen's duality Theorem to establish local stability of certain Strichartz estimates for the kinetic transport equation.
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applications of the funk hecke Theorem to smoothing and Trace estimates
Advances in Mathematics, 2015Co-Authors: Hiroki Saito, Mitsuru SugimotoAbstract:Abstract For a wide class of Kato-smoothing estimates with radial weights, the Funk–Hecke Theorem is used to generate a new expression for the optimal constant in terms of the Fourier transform of the weight, from which several applications are given. For example, we are able to easily establish a unified Theorem, assuming natural power-like asymptotic estimates for the Fourier transform of the weight, from which many well-studied smoothing estimates immediately follow, as well as sharpness of the decay and smoothness exponents. Furthermore, observing that the weight has an everywhere positive Fourier transform in many well-studied cases, our approach allows sharper information regarding the optimal constant and extremisers, substantially extending earlier work of Simon. These observations are very closely related to the Mizohata–Takeuchi conjecture regarding the equivalence of weighted L 2 bounds for the Fourier extension operator on the sphere and the uniform boundedness of the X-ray transform of the weight. For radial weights, this has been independently established by Barcelo–Ruiz–Vega and Carbery–Soria; we provide a short alternative proof in three and higher dimensions of this equivalence when the Fourier transform of the weight is positive, with the optimal relationship between constants. Finally, our approach works for the closely connected Trace Theorems on the sphere where analogous results are given, including the optimal constant and characterisation of extremisers for the inhomogeneous H s ( R d ) → L 2 ( S d − 1 ) Trace Theorem.
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extremisers for the Trace Theorem on the sphere
arXiv: Classical Analysis and ODEs, 2014Co-Authors: Shuji Machihara, Mitsuru SugimotoAbstract:We find all extremisers for the Trace Theorem on the sphere. We also provide a sharp extension for functions belonging to certain Sobolev spaces with angular regularity.