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

Abstract

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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.

How to cite

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Bufetov, 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 -

References

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  2. Alexei Borodin, Grigori Olshanski, Infinite random matrices and ergodic measures, Comm. Math. Phys. 223 (2001), 87-123 Zbl0987.60020MR1860761
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  4. Grigori Olshanski, Unitary representations of infinite-dimensional classical groups (Russian) Zbl1036.43002
  5. Grigori Olshanski, Unitary representations of infinite-dimensional pairs (G,K) and the formalism of R. Howe, Representation of Lie Groups and Related Topics 7 (1990), 165-189, VershikA.M.A. Zbl0724.22020MR1104279
  6. Grigori Olshanski, Anatoli Vershik, Ergodic unitarily invariant measures on the space of infinite Hermitian matrices, Contemporary mathematical physics 175 (1999), 137-175, Amer. Math. Soc., Providence, RI Zbl0853.22016MR1402920
  7. Doug Pickrell, Measures on infinite-dimensional Grassmann manifolds, J. Funct. Anal. 70 (1987), 323-356 Zbl0621.28008MR874060
  8. Doug Pickrell, Mackey analysis of infinite classical motion groups, Pacific J. Math. 150 (1991), 139-166 Zbl0739.22016MR1120717
  9. Marouane Rabaoui, A Bochner type theorem for inductive limits of Gelfand pairs, Ann. Inst. Fourier (Grenoble) 58 (2008), 1551-1573 Zbl1154.22015MR2445827
  10. Marouane Rabaoui, Asymptotic harmonic analysis on the space of square complex matrices, J. Lie Theory 18 (2008), 645-670 Zbl1167.22008MR2493060
  11. Anatoly M. Vershik, A description of invariant measures for actions of certain infinite-dimensional groups, (Russian) Dokl. Akad. Nauk SSSR 218 (1974), 749-752 Zbl0324.28014MR372161

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