A note on free divergence-free vector fields

Hyuga Ito; Akihiro Miyagawa

doi:10.5802/crmath.631

Article de recherche - Analyse fonctionnelle

A note on free divergence-free vector fields
[Note sur les champs de vecteurs libres à divergence nulle]

Hyuga Ito ¹ ; Akihiro Miyagawa ^2,³

¹ Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya, 464-8602, Japan
² Department of Mathematics, Kyoto University, Kitashirakawa Oiwake-cho, Sakyo-ku, 606-8502, Japan
³ Department of Mathematics, University of California San Diego, La Jolla, CA 92093, USA

Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1545-1554.

Abstract (VO)
Résumé

We exhibit an orthonormal basis of cyclic gradients and a (non-orthogonal) basis of the homogeneous free divergence-free vector field on the full Fock space and determine the dimension of Voiculescu’s free divergence-free vector field of degree $k$ or less. Moreover, we also give a concrete formula for the orthogonal projection onto the space of cyclic gradients as well as the free Leray projection.

Nous présentons une base orthonormée de gradients cycliques et une base (non orthogonale) du champ de vecteurs libre homogène à divergence nulle sur l’espace de Fock plein et déterminons la dimension du champ de vecteurs libre au sens de Voiculescu à divergence nulle de degré k ou moins.

En outre, nous donnons une formule concrète pour la projection orthogonale sur l’espace des gradients cycliques ainsi que pour la version libre de la projection de Leray.

Reçu le : 2024-01-22
Révisé le : 2024-03-05
Accepté le : 2024-03-18
Publié le : 2024-11-25

Zbl

DOI : 10.5802/crmath.631

Classification : 46L54, 46L57
Keywords: Free probability, Free semi-circular system, Free divergence-free vector fields, Cyclic gradients
Mots-clés : Probabilité libre, système semi-circulaire libre, champ de vecteurs libre à divergence nulle, gradients cycliques.

Affiliations des auteurs :

Hyuga Ito ¹ ; Akihiro Miyagawa ^{2, 3}

¹ Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya, 464-8602, Japan
² Department of Mathematics, Kyoto University, Kitashirakawa Oiwake-cho, Sakyo-ku, 606-8502, Japan
³ Department of Mathematics, University of California San Diego, La Jolla, CA 92093, USA

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{CRMATH_2024__362_G12_1545_0,
     author = {Hyuga Ito and Akihiro Miyagawa},
     title = {A note on free divergence-free vector fields},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1545--1554},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {362},
     year = {2024},
     doi = {10.5802/crmath.631},
     zbl = {07949971},
     language = {en},
}

TY  - JOUR
AU  - Hyuga Ito
AU  - Akihiro Miyagawa
TI  - A note on free divergence-free vector fields
JO  - Comptes Rendus. Mathématique
PY  - 2024
SP  - 1545
EP  - 1554
VL  - 362
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.631
LA  - en
ID  - CRMATH_2024__362_G12_1545_0
ER  -

%0 Journal Article
%A Hyuga Ito
%A Akihiro Miyagawa
%T A note on free divergence-free vector fields
%J Comptes Rendus. Mathématique
%D 2024
%P 1545-1554
%V 362
%I Académie des sciences, Paris
%R 10.5802/crmath.631
%G en
%F CRMATH_2024__362_G12_1545_0

Hyuga Ito; Akihiro Miyagawa. A note on free divergence-free vector fields. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1545-1554. doi : 10.5802/crmath.631. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.631/

Bibliographie
Cité par

[1] Vladimir I. Arnold; Boris A. Khesin Topological methods in hydrodynamics, Applied Mathematical Sciences, 125, Springer, 2021, xx+455 pages | DOI | MR | Zbl

[2] Hyuga Ito Differential calculus for fully matricial functions I (2023) (https://arxiv.org/abs/2310.09841)

[3] David Jekel; Wuchen Li; Dimitri Shlyakhtenko Tracial smooth functions of non-commuting variables and the free Wasserstein manifold, Diss. Math., Volume 580 (2022), p. 150 | DOI | MR | Zbl

[4] Tobias Mai; Roland Speicher A note on the free and cyclic differential calculus, J. Oper. Theory, Volume 85 (2021) no. 1, pp. 183-215 | DOI | MR | Zbl

[5] Dan Voiculescu A note on cyclic gradients, Indiana Univ. Math. J., Volume 49 (2000) no. 3, pp. 837-841 | DOI | MR | Zbl

[6] Dan Voiculescu Cyclomorphy, Int. Math. Res. Not., Volume 2002 (2002) no. 6, pp. 299-332 | DOI | MR | Zbl

[7] Dan Voiculescu Free entropy, Bull. Lond. Math. Soc., Volume 34 (2002) no. 3, pp. 257-278 | DOI | MR | Zbl

[8] Dan Voiculescu A hydrodynamic exercise in free probability: setting up free Euler equations, Expo. Math., Volume 38 (2020) no. 2, pp. 271-283 | DOI | MR | Zbl

[9] Dan Voiculescu Symmetries of some reduced free product $C^{*}$ -algebras, Operator algebras and their connections with topology and ergodic theory (Buşteni, 1983) (Lecture Notes in Mathematics), Volume 1132, Springer, 1985, pp. 556-588 | DOI | MR | Zbl

[10] Dan Voiculescu The analogues of entropy and of Fisher’s information measure in free probability theory. V. Noncommutative Hilbert transforms, Invent. Math., Volume 132 (1998) no. 1, pp. 189-227 | DOI | MR | Zbl

Cité par Sources :

Commentaires - Politique