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Spectral gap and stability for groups and non-local games

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Název: Spectral gap and stability for groups and non-local games
Autoři: de la Salle, Mikael
Přispěvatelé: Institut Camille Jordan (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS), Équations aux dérivées partielles, analyse (EDPA), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL)
Zdroj: Journal de l’École polytechnique — Mathématiques. 12:1417-1444
Publication Status: Preprint
Informace o vydavateli: Cellule MathDoc/Centre Mersenne, 2025.
Rok vydání: 2025
Témata: Operator Algebras, Artificial intelligence, Classification and Properties of C*-Algebras, [MATH.MATH-OA]Mathematics [math]/Operator Algebras [math.OA], Spectral Theory of Differential Operators, Epistemology, Operator (biology), [MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA], 01 natural sciences, Biochemistry, Gene, [MATH.MATH-GR]Mathematics [math]/Group Theory [math.GR], Quantum Many-Body Systems and Entanglement Dynamics, Machine learning, FOS: Mathematics, 0101 mathematics, Operator Algebras (math.OA), Stability (learning theory), Mathematical Physics, Algebra over a field, Pure mathematics, Spectral Triples, Computer science, Atomic and Molecular Physics, and Optics, FOS: Philosophy, ethics and religion, Philosophy, Chemistry, Physics and Astronomy, Physical Sciences, Simple (philosophy), Repressor, Property (philosophy), Transcription factor, Mathematics, Embedding
Popis: The word ‘stable’ is used to describe a situation when mathematical objects that almost satisfy an equation are close to objects satisfying it exactly. We study operator-algebraic forms of stability for unitary representations of groups and quantum synchronous strategies for non-local games. We observe in particular that simple spectral gap estimates can lead to strong quantitative forms of stability. For example, we prove that the direct product of two (flexibly) Hilbert-Schmidt stable groups is again (flexibly) Hilbert-Schmidt stable, provided that one of them has Kazhdan’s property (T). We also provide a simple form and simple analysis of a non-local game with few questions, with the property that synchronous strategies with large value are close to perfect strategies involving large Pauli matrices. This simplifies one of the steps (the question reduction) in the recent announced resolution of Connes’ embedding problem by Ji, Natarajan, Vidick, Wright and Yuen.
Druh dokumentu: Article
Other literature type
Report
Jazyk: English
ISSN: 2270-518X
DOI: 10.5802/jep.314
DOI: 10.60692/f8pz0-xj224
DOI: 10.60692/k2amw-bwp16
DOI: 10.48550/arxiv.2204.07084
Přístupová URL adresa: http://arxiv.org/abs/2204.07084
Rights: CC BY
Přístupové číslo: edsair.doi.dedup.....a90fbc7728f2b640096b83e506794b8e
Databáze: OpenAIRE
Popis
Abstrakt:The word ‘stable’ is used to describe a situation when mathematical objects that almost satisfy an equation are close to objects satisfying it exactly. We study operator-algebraic forms of stability for unitary representations of groups and quantum synchronous strategies for non-local games. We observe in particular that simple spectral gap estimates can lead to strong quantitative forms of stability. For example, we prove that the direct product of two (flexibly) Hilbert-Schmidt stable groups is again (flexibly) Hilbert-Schmidt stable, provided that one of them has Kazhdan’s property (T). We also provide a simple form and simple analysis of a non-local game with few questions, with the property that synchronous strategies with large value are close to perfect strategies involving large Pauli matrices. This simplifies one of the steps (the question reduction) in the recent announced resolution of Connes’ embedding problem by Ji, Natarajan, Vidick, Wright and Yuen.
ISSN:2270518X
DOI:10.5802/jep.314