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Alexander I. Bufetov
Finiteness of Ergodic Unitarily Invariant Measures on Spaces of Infinite Matrices
(Une mesure sur l’espace des matrices infinies, invariante par l’action du groupe unitaire, doit être finie)
Annales de l'institut Fourier, 64 no. 3 (2014), p. 893-907, doi: 10.5802/aif.2867
Article PDF | Reviews MR 3330157 | Zbl 06387294
Class. Math.: 37A15, 37A25, 28D15, 22E66
Keywords: Infinite-dimensional Lie groups, classification of ergodic measures, Hua-Pickrell measures, orbital measures, weak compactness.

Résumé - Abstract

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.


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