Harmonic Analysis Approach to Gromov–Hausdorff Convergence for Noncommutative Tori

Junge, Marius; Rezvani, Sepideh; Zeng, Qiang

doi:10.1007/s00220-017-3017-4

Cited by 16 publications

(10 citation statements)

References 41 publications

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“…The theory has developed rapidly in further directions over the last decade. Other interesting results and applications can be found in [3,5,19,20,25,31,50,51] and the references therein.…”

Section: Introductionmentioning

confidence: 94%

Spectral multipliers in group algebras and noncommutative Calderón-Zygmund theory

Cadilhac,

Conde-Alonso,

Parcet

2021

Preprint

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In this paper we solve three problems in noncommutative harmonic analysis which are related to endpoint inequalities for singular integrals. In first place, we prove that an L 2 -form of Hörmander's kernel condition suffices for the weak type (1,1) of Calderón-Zygmund operators acting on matrix-valued functions. To that end, we introduce an improved CZ decomposition for martingale filtrations in von Neumann algebras, and apply a very simple unconventional argument which notably avoids pseudolocalization. In second place, we establish as well the weak L 1 endpoint for matrix-valued CZ operators over nondoubling measures of polynomial growth, in the line of the work of Tolsa and Nazarov/Treil/Volberg. The above results are valid for other von Neumann algebras and solve in the positive two open problems formulated in 2009. An even more interesting problem is the lack of L 1 endpoint inequalities for singular Fourier and Schur multipliers over nonabelian groups. Given a locally compact group G equipped with a conditionally negative length ψ : G → R + , we prove that Herz-Schur multipliers with symbol m • ψ satisfying a Mikhlin condition in terms of the ψ-cocycle dimension are of weak type (1, 1). Our result extends to Fourier multipliers for amenable groups and impose sharp regularity conditions on the symbol. The proof crucially combines our new CZ methods with novel forms of recent transference techniques. This L 1 endpoint gives a very much expected inequality which complements the L∞ → BMO estimates proved in 2014 by Junge, Mei and Parcet.

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“…The theory has developed rapidly in further directions over the last decade. Other interesting results and applications can be found in [3,5,19,20,25,31,50,51] and the references therein.…”

Section: Introductionmentioning

confidence: 94%

Spectral multipliers in group algebras and noncommutative Calderón-Zygmund theory

Cadilhac,

Conde-Alonso,

Parcet

2021

Preprint

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“…(See [Lat16a,Lat15], where this was also generalized further, to prove the continuity of the family of quantum 2-tori in Gromov-Hausdorff propinquity, or to allow for more than two unitary generators. For another direction, see [JRZ18], where the authors use a harmonic analysis approach to convergence of the quantum tori with respect to the Gromov-Hausdorff propinquity.) Other examples include approximating spheres with fuzzy spheres (which are finite-dimensional C*-algebras) using the propinquity, as done by Rieffel, which is also a common example from the physics literature [Rie16].…”

Section: Definition 22 a C*-algebramentioning

confidence: 99%

An Ideal Convergence

Aguilar¹,

Brooker²,

Latrémolière³

et al. 2021

Notices Amer. Math. Soc.

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“…Let us illustrate our idea from the perspective of quantum metric spaces. These spaces originate from Connes' work on non-commutative geometry [27] and were later studied by Rieffel [61] and others for their geometric properties such as the Gromov-Hausdorff convergence [41]. A (W * -)quantum metric space (M, L) is given by a von Neumann algebra equipped with a seminorm L : A → [0, ∞) defined on a dense subalgebra A ⊂ M. This semi-norm often arises from a (non-commutative) differential structure, and can be viewed as an abstraction of the notion of Lipschitz constant.…”

Section: Introductionmentioning

confidence: 99%

Ricci curvature of quantum channels on non-commutative transportation metric spaces

Gao,

Rouzé

2021

Preprint

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Following Ollivier's work [57], we introduce the coarse Ricci curvature of a quantum channel as the contraction of non-commutative metrics on the state space. These metrics are defined as a non-commutative transportation cost in the spirit of [38,37], which gives a unified approach to different quantum Wasserstein distances in the literature. We prove that the coarse Ricci curvature lower bound and its dual gradient estimate, under suitable assumptions, imply the Poincaré inequality (spectral gap) as well as transportation cost inequalities. Using intertwining relations, we obtain positive bounds on the coarse Ricci curvature of Gibbs samplers, Bosonic and Fermionic beam-splitters as well as Pauli channels on n-qubits.

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Harmonic Analysis Approach to Gromov–Hausdorff Convergence for Noncommutative Tori

Cited by 16 publications

References 41 publications

Spectral multipliers in group algebras and noncommutative Calderón-Zygmund theory

Spectral multipliers in group algebras and noncommutative Calderón-Zygmund theory

An Ideal Convergence

Ricci curvature of quantum channels on non-commutative transportation metric spaces

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