Metrics on completely positive maps via noncommutative geometry

Are Austad; Erik B\'edos; Jonas Eidesen; Nadia S. Larsen; Tron Omland

arxiv: 2512.10842 · v3 · pith:QPDM6JAZnew · submitted 2025-12-11 · 🧮 math.OA · math.FA· quant-ph

Metrics on completely positive maps via noncommutative geometry

Are Austad , Erik B\'edos , Jonas Eidesen , Nadia S. Larsen , Tron Omland This is my paper

Pith reviewed 2026-05-16 22:58 UTC · model grok-4.3

classification 🧮 math.OA math.FAquant-ph

keywords completely positive mapsmetricsnoncommutative geometryspectral triplesKasparov productsChoi-Jamiołkowski isomorphismC*-algebrasquantum channels

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The pith

Seminorms from noncommutative geometry induce metrics on unital completely positive maps that satisfy stability and chaining.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops methods to induce metrics on unital completely positive maps between C*-algebras by employing seminorms that arise in noncommutative geometry. The central step is the construction of an infinite-dimensional C*-algebraic analogue of the Choi-Jamiołkowski isomorphism, which transfers the seminorms to the space of maps. Under suitable conditions the resulting distances obey the stability and chaining properties that are valued in quantum information theory. The same framework generates the metrics from native noncommutative geometry operations, such as external Kasparov products of spectral triples. A sympathetic reader cares because the work equips the comparison of quantum channels with distances that respect composition and remain well-defined in infinite dimensions.

Core claim

We develop an infinite-dimensional C*-algebraic analogue of the Choi-Jamiołkowski isomorphism. This analogue permits seminorms arising in noncommutative geometry to induce metrics on the set of unital completely positive maps. Under suitable conditions these induced metrics satisfy the quantum information theoretic properties of stability and chaining. Moreover we generate such metrics using constructions native to noncommutative geometry, for example by employing external Kasparov products of spectral triples.

What carries the argument

The infinite-dimensional C*-algebraic analogue of the Choi-Jamiołkowski isomorphism, which maps unital completely positive maps so that seminorms from spectral triples define distances between them.

Load-bearing premise

The infinite-dimensional C*-algebraic analogue of the Choi-Jamiołkowski isomorphism holds and the seminorms from noncommutative geometry induce well-defined metrics on unital completely positive maps.

What would settle it

Computing the induced distance for a concrete pair of unital completely positive maps from a known spectral triple and finding that the distance violates the triangle inequality or the stability property would falsify the claim.

read the original abstract

We study methods of inducing metrics on unital completely positive maps by employing seminorms arising in noncommutative geometry. Our main approach relies on the development of an infinite-dimensional $C^*$-algebraic analogue of the Choi-Jamio\l{}kowski isomorphism. Under suitable conditions, we show that the induced metrics satisfy the quantum information theoretic properties of stability and chaining. Moreover, we show how to generate such metrics using constructions native to noncommutative geometry, by for example using external Kasparov products of spectral triples.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

New infinite-dimensional Choi-Jamiołkowski analogue lets NCG seminorms induce metrics on UCP maps, with stability and chaining under conditions, but point separation needs checking.

read the letter

The main new thing in this paper is the construction of an infinite-dimensional C*-algebraic analogue of the Choi-Jamiołkowski isomorphism. They use this to induce metrics on unital completely positive maps from seminorms coming out of noncommutative geometry. The abstract says that under suitable conditions these metrics satisfy stability and chaining, and that you can generate them using native NCG tools such as external Kasparov products of spectral triples. This is a solid technical step. It gives a concrete way to link geometric data in NCG to properties of CP maps that matter in quantum information. The construction appears to rest on established objects rather than inventing new fitted parameters, which is good. The abstract states the claims clearly enough that a reader can see what is being asserted. The soft spot is whether the induced objects are actually metrics. The stress-test note points out that the finite-dimensional case works because the maximally entangled vector separates points. In infinite dimensions the analogue has to do something different, and if the seminorm does not separate distinct maps, you end up with pseudometrics instead. The paper qualifies the claim with 'under suitable conditions,' so the key question for referees is whether those conditions are spelled out and whether they really guarantee positive distance for different maps. If the separation holds, the rest of the argument should go through without major issues. This work is aimed at people who work in operator algebras and want to bring in ideas from noncommutative geometry or apply them to quantum channels. A reader who needs new examples of metrics on CP maps or who is interested in how spectral triples can be used for this purpose will find something useful here. I would send the paper to peer review. The central construction is new, the claims are checkable, and the potential flaw is specific enough that referees can focus on it without wasting time on vague issues.

Referee Report

1 major / 1 minor

Summary. The paper develops an infinite-dimensional C*-algebraic analogue of the Choi-Jamiołkowski isomorphism to induce metrics on unital completely positive (UCP) maps from seminorms arising in noncommutative geometry. Under suitable conditions the induced objects are claimed to be metrics satisfying stability and chaining; the paper also constructs examples via external Kasparov products of spectral triples.

Significance. If the construction yields genuine metrics (i.e., separates distinct UCP maps) and the stability/chaining properties hold without hidden assumptions, the work would supply a new geometric framework for distances between quantum channels that is native to noncommutative geometry. The explicit use of Kasparov products and spectral triples is a concrete strength that could enable further analytic tools from NCG to be applied in quantum information.

major comments (1)

[Abstract and main construction] The central claim that the induced objects are metrics (rather than pseudometrics) requires that the infinite-dimensional C*-algebraic analogue of the Choi-Jamiołkowski isomorphism separates distinct UCP maps, so that the NCG seminorm is positive on their difference. The abstract invokes “suitable conditions” and external Kasparov products, but the manuscript must explicitly verify that these conditions guarantee d(φ,ψ)>0 whenever φ≠ψ; without such verification the metric property does not follow from the seminorm axioms alone.

minor comments (1)

[Introduction / §2] Clarify the precise definition of the infinite-dimensional analogue isomorphism (including the role of approximate identities or Kasparov modules) and state the exact “suitable conditions” under which stability and chaining are proved.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We appreciate the positive assessment of the potential of our C*-algebraic Choi-Jamiołkowski analogue and Kasparov product constructions for providing a native noncommutative geometry framework for quantum channel distances. We address the single major comment below.

read point-by-point responses

Referee: The central claim that the induced objects are metrics (rather than pseudometrics) requires that the infinite-dimensional C*-algebraic analogue of the Choi-Jamiołkowski isomorphism separates distinct UCP maps, so that the NCG seminorm is positive on their difference. The abstract invokes “suitable conditions” and external Kasparov products, but the manuscript must explicitly verify that these conditions guarantee d(φ,ψ)>0 whenever φ≠ψ; without such verification the metric property does not follow from the seminorm axioms alone.

Authors: We agree that establishing the metric (rather than pseudometric) property requires an explicit verification that the construction separates distinct UCP maps. The suitable conditions in the paper are chosen to ensure that the infinite-dimensional C*-algebraic Choi-Jamiołkowski analogue is faithful (injective) on the relevant space of maps, and that the external Kasparov product with the spectral triple yields a non-degenerate seminorm on the difference. This injectivity follows from the non-degeneracy of the representation and the properties of the Kasparov product, which together imply that the induced seminorm vanishes only when the maps coincide. Nevertheless, we concede that the separation is currently implicit rather than stated as a standalone result. In the revised version we will insert a new proposition immediately after the main construction theorem that explicitly proves: if d(φ,ψ)=0 then φ=ψ, by combining the injectivity of the Choi-Jamiołkowski analogue with the positivity properties of the NCG seminorm. This addition will make the metric property fully rigorous without changing any of the existing theorems or examples. revision: yes

Circularity Check

0 steps flagged

No circularity; construction relies on external NCG objects and stated assumptions

full rationale

The paper develops an infinite-dimensional C*-algebraic analogue of the Choi-Jamiołkowski isomorphism and induces metrics via seminorms from noncommutative geometry, including external Kasparov products of spectral triples. No derivation step reduces by the paper's own equations to a fitted parameter, self-definition, or self-citation chain. The claims of stability and chaining hold under explicitly stated suitable conditions on the analogue and seminorms; these conditions are not shown to be tautological or forced by internal fits. The construction is self-contained against external benchmarks in NCG.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The construction relies on standard properties of C*-algebras, completely positive maps, and noncommutative geometry seminorms. No free parameters, invented entities, or ad-hoc axioms are visible in the abstract.

axioms (2)

domain assumption Existence of an infinite-dimensional C*-algebraic analogue of the Choi-Jamiołkowski isomorphism
Invoked as the main technical tool in the abstract.
domain assumption Seminorms from noncommutative geometry induce metrics on unital completely positive maps
Central step that produces the metrics.

pith-pipeline@v0.9.0 · 5389 in / 1425 out tokens · 25272 ms · 2026-05-16T22:58:09.580027+00:00 · methodology

discussion (0)

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Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/ArithmeticFromLogic.lean Jcost_pos_of_ne_one unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Lipschitz seminorm L_∂(a) = ∥[∂,a]∥... mk_L(ϕ,ψ) := sup{|ϕ(a)−ψ(a)| : L(a)≤1}

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

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math.OA 2026-04 accept novelty 7.0

Extends Bures and quantum metrics to non-unital C*-algebras with faithful traces, proves density space non-compact iff algebra infinite-dimensional, and shows topology comparisons via quantum Lipschitz triples and mat...

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages · cited by 1 Pith paper

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The Podle´ s sphere as a spectral metric space

[AK18] Konrad Aguilar and Jens Kaad. “The Podle´ s sphere as a spectral metric space”. In: J. Geom. Phys. 133 (2018), pp. 260–278. [AJL25] Anshu, David Jekel, and Therese Basa Landry. “Quantum Wasserstein distances for quantum permutation groups”. In: J. Geom. Phys. 217 (2025), Paper No. 105637,

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Quantum metrics on crossed products with groups of polynomial growth

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work page arXiv 2025
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Transmission distance in the space of quantum channels

Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2008, pp. xvi+509. [BLZ23] Diego G. Bussandri, Pedro W. Lamberti, and Karol Zyczkowski. “Transmission distance in the space of quantum channels”. In: Phys. Rev. A 108.1 (2023), Paper No. 012604,

work page 2008
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Matricial Wasserstein-1 distance

[Che+17] Yongxin Chen, Tryphon T. Georgiou, Lipeng Ning, and Allen Tannenbaum. “Matricial Wasserstein-1 distance”. In: IEEE Control Syst. Lett. 1.1 (2017), pp. 14–19. [CR17] Michael Christ and Marc A. Rieffel. “Nilpotent group C ∗-algebras as compact quantum metric spaces”. In: Canad. Math. Bull. 60.1 (2017), pp. 77–94. [Con76] Alain Connes. “Classificati...

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A dual formula for the spectral distance in noncommutative geometry

eprint: arXiv:2505.22519. [DM21] Francesco D’Andrea and Pierre Martinetti. “A dual formula for the spectral distance in noncommutative geometry”. In: J. Geom. Phys. 159 (2021), Paper No. 103920,

work page arXiv 2021
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Quantum Wasserstein distance of order 1 between channels

eprint: arXiv:2511.23121. [DM23] Rocco Duvenhage and Mathumo Mapaya. “Quantum Wasserstein distance of order 1 between channels”. In: Infin. Dimens. Anal. Quantum Probab. Relat. Top. 26.3 (2023), Paper No. 2350006,

work page arXiv 2023
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Balance between quantum Markov semigroups

[DS18] Rocco Duvenhage and Machiel Snyman. “Balance between quantum Markov semigroups”. In: Ann. Henri Poincar´ e19 (2018), pp. 1747–1786. [Far+24] Carla Farsi, Therese Landry, Nadia S. Larsen, and Judith Packer. “Spectral triples for noncommutative solenoids and a Wiener’s lemma”. In: J. Noncommut. Geom. 18.4 (2024), pp. 1415–1452. [GLN05] Alexei Gilchri...

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On spectral triples on crossed products arising from equicontinuous actions

[Haw+13] Andrew Hawkins, Adam Skalski, Stuart White, and Joachim Zacharias. “On spectral triples on crossed products arising from equicontinuous actions”. In: Math. Scand. 113.2 (2013), pp. 262–291. [JKP09] Nathaniel Johnston, David W. Kribs, and Vern I. Paulsen. “Computing stabilized norms for quantum operations via the theory of completely bounded maps”...

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On nonsemisplit extensions, tensor products and exactness of group C∗-algebras

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About the QWEP conjecture

[Oza04] Narutaka Ozawa. “About the QWEP conjecture”. In: Int. J. Math. 15.5 (2004), pp. 501–

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Hyperbolic group C∗-algebras and free-product C∗-algebras as compact quantum metric spaces

[OR05] Narutaka Ozawa and Marc A. Rieffel. “Hyperbolic group C∗-algebras and free-product C∗-algebras as compact quantum metric spaces”. In: Canad. J. Math. 57.5 (2005), pp. 1056–1079. [Pau02] Vern Paulsen. Completely bounded maps and operator algebras . Vol

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A non-nuclear C∗-algebra with the weak expectation property and the local lifting property

Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2002, pp. xii+300. [Pis20a] Gilles Pisier. “A non-nuclear C∗-algebra with the weak expectation property and the local lifting property”. In: Invent. Math. 222.2 (2020), pp. 513–544. [Pis20b] Gilles Pisier. Tensor products of C∗-algebras and operator spaces—the Connes-Kirchbe...

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Metrics on states from actions of compact groups

London Mathematical Society Student Texts. Cambridge University Press, Cambridge, 2020, pp. x+484. [Rie98] Marc A. Rieffel. “Metrics on states from actions of compact groups”. In: Doc. Math. 3 (1998), pp. 215–229. [Rie99] Marc A. Rieffel. “Metrics on state spaces”. In: Doc. Math. 4 (1999), pp. 559–600. [Rie02] Marc A. Rieffel. “Group C∗-algebras as compac...

work page 2020
[14]

Gromov-Hausdorff distance for quantum metric spaces

Contemp. Math. Amer. Math. Soc., Providence, RI, 2004, pp. 315–330. [Rie04b] Marc A. Rieffel. “Gromov-Hausdorff distance for quantum metric spaces”. In: vol

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Matricial bridges for “matrix algebras converge to the sphere

Appendix 1 by Hanfeng Li, Gromov-Hausdorff distance for quantum metric spaces. Matrix algebras converge to the sphere for quantum Gromov-Hausdorff distance. 2004, pp. 1–65. [Rie16] Marc A. Rieffel. “Matricial bridges for “matrix algebras converge to the sphere””. In: Operator algebras and their applications . Vol

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Completely bounded maps between C∗-algebras

Contemp. Math. Amer. Math. Soc., Providence, RI, 2016, pp. 209–233. [Smi83] Roger R. Smith. “Completely bounded maps between C∗-algebras”. In: J. London Math. Soc. (2) 27.1 (1983), pp. 157–166. [Stø13] Erling Størmer. Positive linear maps of operator algebras . Springer Monographs in Mathematics. Springer, Heidelberg, 2013, pp. viii+134. [Val12] Alain Val...

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[17]

On quantum Cayley graphs

url: https://mathoverflow.net/questions/108254/. [Was24] Mateusz Wasilewski. “On quantum Cayley graphs”. In: Doc. Math. 29.no. 6 (2024), pp. 1281–1317. [Wat05] John Watrous. “Notes on super-operator norms induced by Schatten norms”. In: Quan- tum Inf. Comput. 5.1 (2005), pp. 58–68. [Wat09] John Watrous. “Semidefinite programs for completely bounded norms”...

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[1] [1]

The Podle´ s sphere as a spectral metric space

[AK18] Konrad Aguilar and Jens Kaad. “The Podle´ s sphere as a spectral metric space”. In: J. Geom. Phys. 133 (2018), pp. 260–278. [AJL25] Anshu, David Jekel, and Therese Basa Landry. “Quantum Wasserstein distances for quantum permutation groups”. In: J. Geom. Phys. 217 (2025), Paper No. 105637,

work page 2018

[2] [2]

Quantum metrics on crossed products with groups of polynomial growth

eprint: arXiv:2506.04197. [AKK25] Are Austad, Jens Kaad, and David Kyed. “Quantum metrics on crossed products with groups of polynomial growth”. In: Trans. Amer. Math. Soc. 378.3 (2025), pp. 1939–1973. [AK26] Are Austad and David Kyed. “Quantum metrics from length functions on quantum groups”. In: Journal of Functional Analysis 290.4 (2026), p. 111256. [A...

work page arXiv 2025

[3] [3]

Transmission distance in the space of quantum channels

Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2008, pp. xvi+509. [BLZ23] Diego G. Bussandri, Pedro W. Lamberti, and Karol Zyczkowski. “Transmission distance in the space of quantum channels”. In: Phys. Rev. A 108.1 (2023), Paper No. 012604,

work page 2008

[4] [4]

Matricial Wasserstein-1 distance

[Che+17] Yongxin Chen, Tryphon T. Georgiou, Lipeng Ning, and Allen Tannenbaum. “Matricial Wasserstein-1 distance”. In: IEEE Control Syst. Lett. 1.1 (2017), pp. 14–19. [CR17] Michael Christ and Marc A. Rieffel. “Nilpotent group C ∗-algebras as compact quantum metric spaces”. In: Canad. Math. Bull. 60.1 (2017), pp. 77–94. [Con76] Alain Connes. “Classificati...

work page 2017

[5] [5]

A dual formula for the spectral distance in noncommutative geometry

eprint: arXiv:2505.22519. [DM21] Francesco D’Andrea and Pierre Martinetti. “A dual formula for the spectral distance in noncommutative geometry”. In: J. Geom. Phys. 159 (2021), Paper No. 103920,

work page arXiv 2021

[6] [6]

Quantum Wasserstein distance of order 1 between channels

eprint: arXiv:2511.23121. [DM23] Rocco Duvenhage and Mathumo Mapaya. “Quantum Wasserstein distance of order 1 between channels”. In: Infin. Dimens. Anal. Quantum Probab. Relat. Top. 26.3 (2023), Paper No. 2350006,

work page arXiv 2023

[7] [7]

Balance between quantum Markov semigroups

[DS18] Rocco Duvenhage and Machiel Snyman. “Balance between quantum Markov semigroups”. In: Ann. Henri Poincar´ e19 (2018), pp. 1747–1786. [Far+24] Carla Farsi, Therese Landry, Nadia S. Larsen, and Judith Packer. “Spectral triples for noncommutative solenoids and a Wiener’s lemma”. In: J. Noncommut. Geom. 18.4 (2024), pp. 1415–1452. [GLN05] Alexei Gilchri...

work page 2018

[8] [8]

On spectral triples on crossed products arising from equicontinuous actions

[Haw+13] Andrew Hawkins, Adam Skalski, Stuart White, and Joachim Zacharias. “On spectral triples on crossed products arising from equicontinuous actions”. In: Math. Scand. 113.2 (2013), pp. 262–291. [JKP09] Nathaniel Johnston, David W. Kribs, and Vern I. Paulsen. “Computing stabilized norms for quantum operations via the theory of completely bounded maps”...

work page 2013

[9] [9]

On nonsemisplit extensions, tensor products and exactness of group C∗-algebras

Memoirs of the European Mathematical Society. EMS Press, Berlin, 2025, pp. viii+119. [Kir93] Eberhard Kirchberg. “On nonsemisplit extensions, tensor products and exactness of group C∗-algebras”. In: Invent. Math. 112.3 (1993), pp. 449–489. [Kir94] Eberhard Kirchberg. “Discrete groups with Kazhdan’s property T and factorization property are residually fini...

work page 2025

[10] [10]

About the QWEP conjecture

[Oza04] Narutaka Ozawa. “About the QWEP conjecture”. In: Int. J. Math. 15.5 (2004), pp. 501–

work page 2004

[11] [11]

Hyperbolic group C∗-algebras and free-product C∗-algebras as compact quantum metric spaces

[OR05] Narutaka Ozawa and Marc A. Rieffel. “Hyperbolic group C∗-algebras and free-product C∗-algebras as compact quantum metric spaces”. In: Canad. J. Math. 57.5 (2005), pp. 1056–1079. [Pau02] Vern Paulsen. Completely bounded maps and operator algebras . Vol

work page 2005

[12] [12]

A non-nuclear C∗-algebra with the weak expectation property and the local lifting property

Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2002, pp. xii+300. [Pis20a] Gilles Pisier. “A non-nuclear C∗-algebra with the weak expectation property and the local lifting property”. In: Invent. Math. 222.2 (2020), pp. 513–544. [Pis20b] Gilles Pisier. Tensor products of C∗-algebras and operator spaces—the Connes-Kirchbe...

work page 2002

[13] [13]

Metrics on states from actions of compact groups

London Mathematical Society Student Texts. Cambridge University Press, Cambridge, 2020, pp. x+484. [Rie98] Marc A. Rieffel. “Metrics on states from actions of compact groups”. In: Doc. Math. 3 (1998), pp. 215–229. [Rie99] Marc A. Rieffel. “Metrics on state spaces”. In: Doc. Math. 4 (1999), pp. 559–600. [Rie02] Marc A. Rieffel. “Group C∗-algebras as compac...

work page 2020

[14] [14]

Gromov-Hausdorff distance for quantum metric spaces

Contemp. Math. Amer. Math. Soc., Providence, RI, 2004, pp. 315–330. [Rie04b] Marc A. Rieffel. “Gromov-Hausdorff distance for quantum metric spaces”. In: vol

work page 2004

[15] [15]

Matricial bridges for “matrix algebras converge to the sphere

Appendix 1 by Hanfeng Li, Gromov-Hausdorff distance for quantum metric spaces. Matrix algebras converge to the sphere for quantum Gromov-Hausdorff distance. 2004, pp. 1–65. [Rie16] Marc A. Rieffel. “Matricial bridges for “matrix algebras converge to the sphere””. In: Operator algebras and their applications . Vol

work page 2004

[16] [16]

Completely bounded maps between C∗-algebras

Contemp. Math. Amer. Math. Soc., Providence, RI, 2016, pp. 209–233. [Smi83] Roger R. Smith. “Completely bounded maps between C∗-algebras”. In: J. London Math. Soc. (2) 27.1 (1983), pp. 157–166. [Stø13] Erling Størmer. Positive linear maps of operator algebras . Springer Monographs in Mathematics. Springer, Heidelberg, 2013, pp. viii+134. [Val12] Alain Val...

work page 2016

[17] [17]

On quantum Cayley graphs

url: https://mathoverflow.net/questions/108254/. [Was24] Mateusz Wasilewski. “On quantum Cayley graphs”. In: Doc. Math. 29.no. 6 (2024), pp. 1281–1317. [Wat05] John Watrous. “Notes on super-operator norms induced by Schatten norms”. In: Quan- tum Inf. Comput. 5.1 (2005), pp. 58–68. [Wat09] John Watrous. “Semidefinite programs for completely bounded norms”...

work page 2024