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Frederic Latremoliere

Publications and source records attributed to Frederic Latremoliere.

At least 19 recordsLinked to original sources
arXiv2026

How to approximate the flat spectral triple of a quantum torus by fuzzy tori : a twisted tale

We prove that the classical and the quantum flat torus can be rigorously approximated at a differential level by finite-dimensional fuzzy tori within the framework of the spectral propinquity. Standard attempts to establish this convergence are traditionally obstructed by the intrinsic non-locality of discrete calculus and the subsequent failure of the Leibniz rule. While contemporary alternatives such as spectral truncations circumvent this issue by abandoning $C^*$-algebras in favor of operator systems, we instead preserve the $C^*$-algebraic category by generalizing the commutator formula. To this end, we introduce a relaxed notion of a twisted spectral triple where the twist is a linear map acting as a discretized Riesz transform that encapsulates the non-locality of the discrete world. Self-adjointness is obtained by adjoining the backward difference to the forward difference and representing this doubled calculus with the creation and annihilation operators of the CAR algebra. We also require the L-seminorm to control both the twisted commutator and the displacement of the twist from the identity. By extending the spectral propinquity to this generalized setting of twisted spectral triples with possibly unbounded twists, we prove that fuzzy tori equipped with their natural two-sided discrete calculus converge to an amplification of the standard flat Dirac triple, while the underlying twists converge to the identity.

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arXiv2026

Spectral continuity of almost commutative manifolds for the $C^1$ topology on Riemannian metrics

Almost commutative models provide a framework for Connes' work on the standard model of particle physics. These models are constructed as products of a the canonical spectral triple of a compact connected spin manifold with a finite dimensional spectral triple. Motivated by the fundamental question of the dependence of the spectra of Dirac operators under change of metrics, we prove the continuity of the spectra of Dirac operators for almost commutative models as functions of the underlying Riemannian metric. We allow both the Riemannian metric (in the $C^1$ topology) and the Dirac operator of the finite-dimensional factor to vary simultaneously. Since the physics of the system is fundamentally encoded in this spectrum, this result is a form of stability result regarding the geometry, or physical, content of these models. This result is based upon a novel approach to prove continuity of spectra of Dirac operators using the spectral propinquity. Notably, this method provides a new, different proof of the classical results as well. To illustrate the versatility of our new method, we also apply our results to completely non-commutative family of examples, including quantum tori and quantum solenoids.

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arXiv2025

The quantum Gromov-Hausdorff Hypertopology on the class of pointed Proper Quantum Metric Spaces

We introduce a hypertopology, induced by an inframetric up to full quantum isometry, on the class of pointed proper quantum metric spaces, which are separable, possibly non-unital, C*-algebras endowed with an analogue of the Lipschitz seminorm, with a distinguished state, and with a particular type of approximate units. Our hypertopology provides an analogue of the Gromov-Hausdorff distance on proper metric spaces, and in fact, convergence in the latter implies convergence in the former. Moreover, when restricted to the class of quantum compact metric spaces, our new topology is compatible with the topology of the Gromov-Hausdorff propinquity. We include new examples of noncompact, noncommutative pointed proper quantum metric spaces which are limits, for our new topology, of finite dimensional quantum compact metric spaces. This article thus provides a first answer to the challenging question of how to extend noncommutative metric geometry to the locally compact quantum space realm.

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arXiv2025

Collapse in Noncommutative Geometry and Spectral Continuity

If two compact quantum metric spaces are close in the metric sense, then how similar are they, as noncommutative spaces? In the classical realm of Riemannian geometry, informally, if two manifolds are close in the Gromov-Hausdorff distance, and belong to a class of manifolds with bounded curvature and diameter, then the spectra of their Laplacian or Dirac operators are also close under many scenari. Of particular interest is the case where a sequence of manifolds converge for the Gromov-Hausdorff distance to a manifold of lower dimension, and the question of the continuity, in some sense, of the spectra of geometrically relevant operators. In this paper, we initiate the study of the continuity of spectra and other properties of metric spectral triples under collapse in the noncommutative realm. As a first step in this study, we work with collapse for the spectral propinquity, an analogue of the Gromov-Hausdorff distance for spectral triples introduced by the second author, i.e. a form of metric for differential structures. Inspired by results from collapse in Riemannian geometry, we begin with the study of spectral triples which decompose, in some sense, in a vertical and a horizontal direction, and we collapse these spectral triples along the vertical direction. We obtain convergence results, and by the work of the second author, we conclude continuity results for the spectra of the Dirac operators of these spectral triples. Examples include collapse of product of spectral triples with one Abelian factor, $U(1)$ principal bundles over Riemannian spin manifolds, and noncommutative principal bundles, including C*-crossed-products and other noncommutative bundles.

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arXiv2024

Spectral Triples on noncommutative solenoids from the standard spectral triples on quantum tori

We address the natural question: as noncommutative solenoids are inductive limits of quantum tori, do the standard spectral triples on quantum tori converge to some spectral triple on noncommutative solenoid for the spectral propinquity? We answer this question by showing that, using appropriate bounded perturbation of the spectral triples on quantum tori, such a spectral triple on noncommutative solenoid can be constructed.

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arXiv2024

Domains of quantum metrics on AF algebras

Given a compact quantum metric space (A, L), we prove that the domain of L coincides with A if and only if A is finite dimensional. We then show how one can explicitly build many quantum metrics with distinct domains on infinite-dimensional AF algebras. In the last section, we provide a strategy for calculating the distance between certain states in these quantum metrics, which allow us to calculate the distance between pure states in these quantum metrics on the quantized interval and on the Cantor space.

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arXiv2023

Continuity of the Spectrum of Dirac Operators of Spectral Triples for the Spectral Propinquity

The spectral propinquity is a distance, up to unitary equivalence, on the class of metric spectral triples. We prove in this paper that if a sequence of metric spectral triples converges for the propinquity, then the spectra of the Dirac operators for these triples do converge to the spectrum of the Dirac operator at the limit. We obtain this result by first proving that, in an appropriate sense induced by some natural metric, the bounded, continuous functional calculi defined by the Dirac operators also converge. As an application of our work, we see, in particular, that action functionals of a wide class of metric spectral triples are continuous for the spectral propinquity, which clearly connects convergence for the spectral propinquity with the applications of noncommutative geometry to mathematical physics. This fact is a consequence of results on the continuity of the multiplicity of eigenvalues of Dirac operators. In particular, we formalize convergence of adjoinable operators of different C*-correspondences, endowed with appropriate quantum metric data.

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arXiv2023

Isometry groups of inductive limits of metric spectral triples and Gromov-Hausdorff convergence

In this paper we study the groups of isometries and the set of bi-Lipschitz automorphisms of spectral triples from a metric viewpoint, in the propinquity framework of Latremoliere. In particular we prove that these groups and sets are compact in the automorphism group of the spectral triple C*-algebra with respect to the Monge-Kantorovich metric, which induces the topology of pointwise convergence. We then prove a necessary and sufficient condition for the convergence of the actions of various groups of isometries, in the sense of the covariant version of the Gromov-Hausdorff propinquity -- a noncommutative analogue of the Gromov-Hausdorff distance -- when working in the context of inductive limits of quantum compact metric spaces and metric spectral triples. We illustrate our work with examples including AF algebras and noncommutative solenoids.

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arXiv2023

The strongly Leibniz property and the Gromov--Hausdorff propinquity

We construct a new version of the dual Gromov--Hausdorff propinquity that is sensitive to the strongly Leibniz property. In particular, this new distance is complete on the class of strongly Leibniz quantum compact metric spaces. Then, given an inductive limit of C*-algebras for which each C*-algebra of the inductive limit is equipped with a strongly Leibniz L-seminorm, we provide sufficient conditions for placing a strongly Leibniz L-seminorm on an inductive limit such that the inductive sequence converges to the inductive limit in this new Gromov--Hausdorff propinquity. As an application, we place new strongly Leibniz L-seminorms on AF-algebras using Frobenius--Rieffel norms, for which we have convergence of the Effros--Shen algebras in the Gromov--Hausdorff propinquity with respect to their irrational parameter.

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arXiv2022

Convergence of inductive sequences of spectral triples for the spectral propinquity

In the context of metric geometry, we introduce a new necessary and sufficient condition for the convergence of an inductive sequence of quantum compact metric spaces for the Gromov-Hausdorff propinquity, which is a noncommutative analogue of the Gromov-Hausdorff distance for compact metric spaces. This condition is easy to verify in many examples, such as quantum compact metric spaces associated to AF algebras or certain twisted convolution C*-algebras of discrete inductive limit groups. Our condition also implies the convergence of an inductive sequence of spectral triples in the sense of the spectral propinquity, a generalization of the Gromov-Hausdorff propinquity on quantum compact metric spaces to the space of metric spectral triples. In particular we show the convergence of the state spaces of the underlying C*-algebras as quantum compact metric spaces, and also the convergence of the quantum dynamics induced by the Dirac operators in the spectral triples. We apply these results to new classes of inductive limit of even spectral triples on noncommutative solenoids and Bunce-Deddens C*-algebras. Our construction, which involves length functions with bounded doubling, adds geometric information and highlights the structure of these twisted C*-algebras as inductive limits.

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arXiv2022

The Gromov-Hausdorff propinquity for metric Spectral Triples

We define a metric on the class of metric spectral triples, which is null exactly between spectral triples with unitary equivalent Dirac operators and *-isomorphic underlying C*-algebras. This metric dominates the propinquity, and thus implies metric convergence of the compact quantum metric spaces induced by metric spectral triples. In the process of our construction, we also introduce the covariant modular propinquity, as a key component for the definition of the spectral propinquity.

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arXiv2021

Quantum Metric Spaces and the Gromov-Hausdorff Propinquity

We present a survey of the dual Gromov-Hausdorff propinquity, a noncommutative analogue of the Gromov-Hausdorff distance which we introduced to provide a framework for the study of the noncommutative metric properties of C*-algebras. We first review the notions of quantum locally compact metric spaces, and present various examples of such structures. We then explain the construction of the dual Gromov-Hausdorff propinquity, first in the context of quasi-Leibniz quantum compact metric spaces, and then in the context of pointed quantum proper metric spaces. We include a few new result concerning perturbations of the metrics on Leibniz quantum compact metric spaces in relation with the dual Gromov-Hausdorff propinquity.

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arXiv2021

Equivalence of Quantum Metrics with a common domain

We characterize Lipschitz morphisms between quantum compact metric spaces as those *-morphisms which preserve the domain of certain noncommutative analogues of Lipschitz seminorms, namely lower semi-continuous Lip-norms. As a corollary, lower semi-continuous Lip-norms with a shared domain are in fact equivalent. We then note that when a family of lower semi-continuous Lip-norms are uniformly equivalent, then they give rise to totally bounded classes of quantum compact metric spaces, and we apply this observation to several examples of perturbations of quantum metric spaces. We also construct the noncommutative generalization of the Lipschitz distance between quantum compact metric spaces.

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arXiv2021

Convergence of Spectral Triples on Fuzzy Tori to Spectral Triples on Quantum Tori

Fuzzy tori are finite dimensional C*-algebras endowed with an appropriate notion of noncommutative geometry inherited from an ergodic action of a finite closed subgroup of the torus, which are meant as finite dimensional approximations of tori and more generally, quantum tori. A mean to specify the geometry of a noncommutative space is by constructing over it a spectral triple. We prove in this paper that we can construct spectral triples on fuzzy tori which, as the dimension grow to infinity and under other natural conditions, converge to a natural spectral triple on quantum tori, in the sense of the spectral propinquity. This provides a formal assertion that indeed, fuzzy tori approximate quantum tori, not only as quantum metric spaces, but as noncommutative differentiable manifolds -- including convergence of the state spaces as metric spaces and of the quantum dynamics generated by the Dirac operators of the spectral triples, in an appropriate sense.

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arXiv2021

Metric Approximations of Spectral Triples on the Sierpiński Gasket and other fractal curves

Noncommutative geometry provides a framework, via the construction of spectral triples, for the study of the geometry of certain classes of fractals. Many fractals are constructed as natural limits of certain sets with a simpler structure: for instance, the Sierpiński is the limit of finite graphs consisting of various affine images of an equilateral triangle. It is thus natural to ask whether the spectral triples, constructed on a class of fractals called piecewise $C^1$-fractal curves, are indeed limits, in an appropriate sense, of spectral triples on the approximating sets. We answer this question affirmatively in this paper, where we use the spectral propinquity on the class of metric spectral triples, in order to formalize the sought-after convergence of spectral triples. Our results and methods are relevant to the study of analysis on fractals and have potential physical applications.

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arXiv2020

Bunce-Deddens algebras as quantum Gromov-Hausorff distance limits of circle algebras

We show that Bunce-Deddens algebras, which are AT-algebras, are also limits of circle algebras for Rieffel's quantum Gromov-Hausdorff distance, and moreover, form a continuous family indexed by the Baire space. To this end, we endow Bunce-Deddens algebras with a quantum metric structure, a step which requires that we reconcile the constructions of the Latremoliere's Gromov-Hausdorff propinquity and Rieffel's quantum Gromov-Hausdorff distance when working on order-unit quantum metric spaces. This work thus continues the study of the connection between inductive limits and metric limits.

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arXiv2020

A survey of the Preservation of Symmetries by the Dual Gromov-Hausdorff Propinquity

We survey the symmetry preserving properties for the dual propinquity, under natural non-degeneracy and equicontinuity conditions. These properties are best formulated using the notion of the covariant propinquity when the symmetries are encoded via the actions of proper monoids and groups. We explore the issue of convergence of Cauchy sequences for the covariant propinquity, which captures, via a compactness result, the fact that proper monoid actions can pass to the limit for the dual propinquity.

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