Groups of isometries of the Cuntz algebras

Conti, Roberto; Rossi, Stefano

doi:10.4171/dm/856

Cited by 4 publications

(4 citation statements)

References 11 publications

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“…Other comparison results concerning the two isometry groups for the case of Cuntz algebras have been obtained in [6].…”

Section: Definition 22 ([14]mentioning

confidence: 99%

“…From the point of view of non-commutative geometry, a natural problem is to understand what is the right notion of isometry for a non-commutative manifold. Such topic was investigated in [14,15,16,13,6] and at the time being the two definitions of noncommutative isometry appearing in these manuscripts are the only natural candidates known to the authors. More precisely, these are the automorphisms implemented by unitaries commuting with the Dirac operator and the automorphisms leading to preservation of the Connes distance on the state space, giving rise to the two groups Iso and ISO, respectively.…”

Section: Introductionmentioning

confidence: 99%

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On isometries of spectral triples associated to $AF$-algebras and crossed products

Bassi¹,

Conti²

2023

Preprint

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We examine the structure of two possible candidates of isometry groups for the spectral triples on AF -algebras introduced by Christensen and Ivan. In particular, we completely determine the isometry group introduced by Park, and observe that these groups coincide in the case of the Cantor set. We also show that the construction of spectral triples on crossed products given by Hawkins, Skalski, White and Zacharias, is suitable for the purpose of lifting isometries.

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“…Other comparison results concerning the two isometry groups for the case of Cuntz algebras have been obtained in [6].…”

Section: Definition 22 ([14]mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On isometries of spectral triples associated to $AF$-algebras and crossed products

Bassi¹,

Conti²

2023

Preprint

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“…This work requires us to first prove that the group of isometries of a quantum compact metric space is compact. For the Riemannian manifold case, noncommutative tori and other examples, Iso was studied by Park in [31,32], for spectral triples over twisted reduced group C * -algebras associated to a length function as defined in [9] by Long-Wu in [30], for Goffeng-Mesland [15] spectral triples on Cuntz algebras by Conti-Rossi in [11], and for the Christensen-Ivan [6] spectral triples of AF-algebras by Bassi-Conti in [2]. Conti-Farsi consider both Iso and ISO for Kellendonk-Savinien spectral triples in [10].…”

Section: Introductionmentioning

confidence: 99%

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

Bassi¹,

Conti²,

Farsi³

et al. 2023

Preprint

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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 Latrémolière. 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. CONTENTS 1. Introduction 1 Acknowledgments 5 2. The Isometry Groups of a Metric Spectral Triple 5 2.1. The Isometry Group of a Quantum Compact Metric Space 6 2.2. The Isometry Group of a Spectral Triple 12 3. The Propinquity and Isometry Groups 17 3.1. The Covariant Propinquity 17 3.2. Covariant Bridge Builders 19 4. Applications 29 4.1. The Identity as a Bridge Builder 30 4.2. Almost Inners 31 4.3. AF Algebras with Faithful Traces 32 4.4. Dual Actions and Inductive Limits 35 References 37

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“…This work requires us to first prove that the group of isometries of a quantum compact metric space is compact. For the Riemannian manifold case, noncommutative tori and other examples,

{\mathsf {Iso}}

was studied by Park in [31, 32], for spectral triples over twisted reduced group

C^*

‐algebras associated to a length function as defined in [9] by Long–Wu in [30], for Goffeng–Mesland [15] spectral triples on Cuntz algebras by Conti‐Rossi in [11], and for the Christensen–Ivan [6] spectral triples of AF‐algebras by Bassi–Conti in [2]. Conti–Farsi consider both

{\mathsf {Iso}}

and

{\mathsf {ISO}}

for Kellendonk–Savinien spectral triples in [10].…”

Section: Introductionmentioning

confidence: 99%

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

Bassi

Conti

Farsi

et al. 2023

Journal of London Math Soc

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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 Latrémolière. In particular, we prove that these groups and sets are compact in the automorphism group of the spectral triple ‐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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Groups of isometries of the Cuntz algebras

Abstract: We provide a new interpretation of the group of Bogolubov automorphisms of the Cuntz algebras O n and the group Aut(O n , F n ) of all automorphisms preserving the UHF subalgebra F n ⊂ O n as the isometry groups coming from two distinct spectral triples on O n .

Cited by 4 publications

References 11 publications

On isometries of spectral triples associated to $AF$-algebras and crossed products

On isometries of spectral triples associated to $AF$-algebras and crossed products

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

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

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