Finiteness of Ergodic Unitarily Invariant Measures on Spaces of Infinite Matrices
Annales de l'Institut Fourier, Volume 64 (2014) no. 3, pp. 893-907.

The main result of this note, Theorem 1.3, is the following: a Borel measure on the space of infinite Hermitian matrices, that is invariant and ergodic under the action of the infinite unitary group and that admits well-defined projections onto the quotient space of “corners" of finite size, must be finite. A similar result, Theorem 1.1, is also established for unitarily invariant measures on the space of all infinite complex matrices. These results imply that the infinite Hua-Pickrell measures of Borodin and Olshanski have finite ergodic components.

The proof is based on the approach of Olshanski and Vershik. First, it is shown that if the sequence of orbital measures assigned to almost every point is weakly precompact, then our ergodic measure must indeed be finite. The second step, which completes the proof, shows that if a unitarily-invariant measure admits well-defined projections onto the quotient space of finite corners, then for almost every point the corresponding sequence of orbital measures is indeed weakly precompact.

Le résultat principal de cet article, Théorème 1.3, affirme que si une mesure borélienne sur l’espace des matrices hermitiennes infinies, invariante et ergodique par l’action du groupe unitaire infini admet, en plus, des projections sur l’espace quotient des matrices finies, alors la mesure est elle-même finie. Un résultat similaire, Théorème 1.1, est obtenu pour les mesures invariantes par l’action du groupe unitaire sur l’espace de toutes les matrices complexes infinies. Ces résultats impliquent que toutes les composantes ergodiques des mesures infinies de Hua-Pickrell introduites par Borodin et Olshanski doivent être finies.

L’argument se base sur l’approche d’Olshanski et Vershik. On démontre d’abord que la mesure ergodique doit être finie si la suite des mesures orbitales d’un point générique est précompacte. Le deuxième pas qui conclut la preuve est la vérification de la précompacité des suites des mesures orbitales.

DOI: 10.5802/aif.2867
Classification: 37A15, 37A25, 28D15, 22E66
Keywords: Infinite-dimensional Lie groups, classification of ergodic measures, Hua-Pickrell measures, orbital measures, weak compactness.
Mot clés : Groupes de Lie de dimension infinie, classification des mesures ergodiques, mesures de Hua-Pickrell, mesures orbitales, compacité étroite.
Bufetov, Alexander I. 1, 2, 3, 4, 5

1 Aix-Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373, Marseille, France
2 The Steklov Institute of Mathematics, Moscow
3 The Institute for Information Transmission Problems, Moscow
4 National Research University Higher School of Economics, Moscow
5 Rice University, Houston
@article{AIF_2014__64_3_893_0,
     author = {Bufetov, Alexander  I.},
     title = {Finiteness of {Ergodic} {Unitarily} {Invariant} {Measures} on {Spaces} of {Infinite} {Matrices}},
     journal = {Annales de l'Institut Fourier},
     pages = {893--907},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {64},
     number = {3},
     year = {2014},
     doi = {10.5802/aif.2867},
     zbl = {06387294},
     mrnumber = {3330157},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.2867/}
}
TY  - JOUR
AU  - Bufetov, Alexander  I.
TI  - Finiteness of Ergodic Unitarily Invariant Measures on Spaces of Infinite Matrices
JO  - Annales de l'Institut Fourier
PY  - 2014
SP  - 893
EP  - 907
VL  - 64
IS  - 3
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.2867/
DO  - 10.5802/aif.2867
LA  - en
ID  - AIF_2014__64_3_893_0
ER  - 
%0 Journal Article
%A Bufetov, Alexander  I.
%T Finiteness of Ergodic Unitarily Invariant Measures on Spaces of Infinite Matrices
%J Annales de l'Institut Fourier
%D 2014
%P 893-907
%V 64
%N 3
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.2867/
%R 10.5802/aif.2867
%G en
%F AIF_2014__64_3_893_0
Bufetov, Alexander  I. Finiteness of Ergodic Unitarily Invariant Measures on Spaces of Infinite Matrices. Annales de l'Institut Fourier, Volume 64 (2014) no. 3, pp. 893-907. doi : 10.5802/aif.2867. http://www.numdam.org/articles/10.5802/aif.2867/

[1] Bogachev, Vladimir I. Measure theory, II, Springer-Verlag, Berlin, 2007 | MR | Zbl

[2] Borodin, Alexei; Olshanski, Grigori Infinite random matrices and ergodic measures, Comm. Math. Phys., Volume 223 (2001) no. 1, pp. 87-123 | MR | Zbl

[3] Bufetov, Alexander I. Ergodic decomposition for measures quasi-invariant under Borel actions of inductively compact groups, Sbornik Mathematics, Volume 205 (2014) no. 2, pp. 39-71 | MR

[4] Olshanski, Grigori Unitary representations of infinite-dimensional classical groups (Russian) (D.Sci Thesis, Institute of Geography of the Russian Academy of Sciences; online at http://www.iitp.ru/upload/userpage/52/Olshanski_thesis.pdf)

[5] Olshanski, Grigori; Vershik, A.M.; Zhelobenko, D. P. Unitary representations of infinite-dimensional pairs (G,K) and the formalism of R. Howe, Representation of Lie Groups and Related Topics (Advanced Studies in Contemporary Mathematics), Volume 7, Gordon and Breach, 1990, pp. 165-189 (online at http://www.iitp.ru/upload/userpage/52/HoweForm.pdf) | MR | Zbl

[6] Olshanski, Grigori; Vershik, Anatoli Ergodic unitarily invariant measures on the space of infinite Hermitian matrices, Contemporary mathematical physics (Amer. Math. Soc. Transl. Ser. 2), Volume 175, Amer. Math. Soc., Providence, RI, 1999, pp. 137-175 | MR | Zbl

[7] Pickrell, Doug Measures on infinite-dimensional Grassmann manifolds, J. Funct. Anal., Volume 70 (1987), pp. 323-356 | MR | Zbl

[8] Pickrell, Doug Mackey analysis of infinite classical motion groups, Pacific J. Math., Volume 150 (1991) no. 1, pp. 139-166 | MR | Zbl

[9] Rabaoui, Marouane A Bochner type theorem for inductive limits of Gelfand pairs, Ann. Inst. Fourier (Grenoble), Volume 58 (2008) no. 5, pp. 1551-1573 | Numdam | MR | Zbl

[10] Rabaoui, Marouane Asymptotic harmonic analysis on the space of square complex matrices, J. Lie Theory, Volume 18 (2008) no. 3, pp. 645-670 | MR | Zbl

[11] Vershik, Anatoly M. A description of invariant measures for actions of certain infinite-dimensional groups, (Russian) Dokl. Akad. Nauk SSSR, Volume 218 (1974), pp. 749-752 | MR | Zbl

Cited by Sources: