# Finiteness of Ergodic Unitarily Invariant Measures on Spaces of Infinite Matrices

Alexander I. Bufetov^{[1]}

- [1] Aix-Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373, Marseille, France The Steklov Institute of Mathematics, Moscow The Institute for Information Transmission Problems, Moscow National Research University Higher School of Economics, Moscow Rice University, Houston

Annales de l’institut Fourier (2014)

- Volume: 64, Issue: 3, page 893-907
- ISSN: 0373-0956

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topBufetov, Alexander I.. "Finiteness of Ergodic Unitarily Invariant Measures on Spaces of Infinite Matrices." Annales de l’institut Fourier 64.3 (2014): 893-907. <http://eudml.org/doc/275466>.

@article{Bufetov2014,

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.},

affiliation = {Aix-Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373, Marseille, France The Steklov Institute of Mathematics, Moscow The Institute for Information Transmission Problems, Moscow National Research University Higher School of Economics, Moscow Rice University, Houston},

author = {Bufetov, Alexander I.},

journal = {Annales de l’institut Fourier},

keywords = {Infinite-dimensional Lie groups; classification of ergodic measures; Hua-Pickrell measures; orbital measures; weak compactness; infinite-dimensional Lie groups},

language = {eng},

number = {3},

pages = {893-907},

publisher = {Association des Annales de l’institut Fourier},

title = {Finiteness of Ergodic Unitarily Invariant Measures on Spaces of Infinite Matrices},

url = {http://eudml.org/doc/275466},

volume = {64},

year = {2014},

}

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

PB - Association des Annales de l’institut Fourier

VL - 64

IS - 3

SP - 893

EP - 907

AB - 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.

LA - eng

KW - Infinite-dimensional Lie groups; classification of ergodic measures; Hua-Pickrell measures; orbital measures; weak compactness; infinite-dimensional Lie groups

UR - http://eudml.org/doc/275466

ER -

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