Asymptotic Tensor Powers of Banach Spaces

Aubrun, Guillaume; Müller-Hermes, Alexander

doi:10.5802/aif.3736

Asymptotic Tensor Powers of Banach Spaces
[Puissances tensorielles asymptotiques d’espaces de Banach]

Aubrun, Guillaume ¹ ; Müller-Hermes, Alexander ^1,²

¹ Institut Camille Jordan, Université Claude Bernard Lyon 1, 43 boulevard du 11 novembre 1918, 69622 Villeurbanne cedex (France)
² Department of Mathematics, University of Oslo, P.O. Box 1053, Blindern, 0316 Oslo(Norway)

Annales de l'Institut Fourier, Online first, 28 p.

Abstract (VO)
Résumé

We study the asymptotic behaviour of large tensor powers of normed spaces and of operators between them. We define the tensor radius of a finite-dimensional normed space $X$ as the limit of the sequence $A_{k}^{1 / k}$ , where $A_{k}$ is the equivalence constant between the projective and injective norms on $X^{\otimes k}$ . We show that Euclidean spaces are characterized by the property that their tensor radius equals their dimension. Moreover, we compute the tensor radius for spaces with enough symmetries, such as the spaces $ℓ_{p}^{n}$ . We also define the tensor radius of an operator $T$ as the limit of the sequence $B_{k}^{1 / k}$ , where $B_{k}$ is the injective-to-projective norm of $T^{\otimes k}$ . We show that the tensor radius of an operator whose domain or range is Euclidean is equal to its nuclear norm, and give some evidence that this property might characterize Euclidean spaces.

Nous étudions le comportement asymptotique des grandes puissances tensorielles des espaces normés et de leurs opérateurs. Nous définissons le rayon tensoriel d’un espace normé $X$ de dimension finie comme la limite de la suite $A_{k}^{1 / k}$ , où $A_{k}$ est la constante d’équivalence entre les normes injective et projective sur $X^{\otimes k}$ . Nous montrons que les espaces euclidiens sont caractérisés par le fait que leur rayon tensoriel est égal à leur dimension. De plus, nous calculons le rayon tensoriel des espaces ayant sufisamment de symétries, comme les espaces $ℓ_{p}^{n}$ . Nous définissons également le rayon tensoriel d’un opérateur $T$ comme la limite de la suite $B_{k}^{1 / k}$ , où $B_{k}$ est la norme injective-vers-projective de $T^{\otimes k}$ . Nous montrons que le rayon tensoriel d’un opérateur défini sur un espace euclidien ou à valeurs dans un espace euclidien est égal à sa norme nucléaire, et suggérons que cette propriété pourrait caractériser les espaces euclidiens.

Reçu le : 2023-01-16
Révisé le : 2023-08-08
Accepté le : 2023-12-21
Première publication : 2025-09-22

DOI : 10.5802/aif.3736

Classification : 46B28, 46B20, 47A80, 52A21
Keywords: tensor radius, projective and injective norms, Banach–Mazur distance, tensor powers
Mots-clés : rayon tensoriel, normes injective et projective, distance de Banach–Mazur, puissances tensorielles

Affiliations des auteurs :

Aubrun, Guillaume ¹ ; Müller-Hermes, Alexander ^{1, 2}

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     author = {Aubrun, Guillaume and M\"uller-Hermes, Alexander},
     title = {Asymptotic {Tensor} {Powers} of {Banach} {Spaces}},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     year = {2025},
     doi = {10.5802/aif.3736},
     language = {en},
     note = {Online first},
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Aubrun, Guillaume; Müller-Hermes, Alexander. Asymptotic Tensor Powers of Banach Spaces. Annales de l'Institut Fourier, Online first, 28 p.

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