Finiteness of Ergodic Unitarily Invariant Measures on Spaces of Infinite Matrices

Bufetov, Alexander I.

Annales de l'Institut Fourier, Tome 64 (2014) no. 3, pp. 893-907.

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.

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.

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.

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