2011/08/12 by Alexander I. Bufetov, Bufetov, Alexander I.
Mathematics · #22F10 #28D15 #37A15 #Advanced Algebra and Geometry #Dynamical Systems (math.DS) #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Dynamics and Fractals #Random Matrices and Applications #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.1108.2737
openalex publication_date 2011/08/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The main result of this note, Theorem 2, is the following: a Borel measure on\nthe space of infinite Hermitian matrices, that is invariant under the action of\nthe infinite unitary group and that admits well-defined projections onto the\nquotient space of "corners" of finite size, must be finite. A similar result,\nTheorem 1, is also established for unitarily invariant measures on the space of\nall infinite complex matrices. These results, combined with the ergodic\ndecomposition theorem of [3], imply that the infinite Hua-Pickrell measures of\nBorodin and Olshanski [2] have finite ergodic components.\n The proof is based on the approach of Olshanski and Vershik [6]. First, it is\nshown that if the sequence of orbital measures assigned to almost every point\nis weakly precompact, then our ergodic measure must indeed be finite. The\nsecond step, which completes the proof, shows that if a unitarily-invariant\nmeasure admits well-defined projections onto the quotient space of finite\ncorners, then for almost every point the corresponing sequence of orbital\nmeasures is indeed weakly precompact.\n