Bufetov, Alexander I. 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.) (English. French summary) Zbl 1322.37002 Ann. Inst. Fourier 64, No. 3, 893-907 (2014). Let \(\mathrm{Mat}(\mathbb{N}, \mathbb{C})\) be the space of all infinite matrices with complex entries, and \(\mathrm{U}(\infty)\) be the infinite unitary group. The group \(\mathrm{U}(\infty)\times \mathrm{U}(\infty)\) acts on \(\mathrm{Mat}(\mathbb{N},\mathbb{C})\) by multiplication on both sides.For \(m\in \mathbb{N}\) denote by \(\mathfrak{F}(m;\mathrm{Mat}(\mathbb{N},\mathbb{C}))\) the space of Borel measures \(\nu\) on \(\mathrm{Mat}(\mathbb{N},\mathbb{C})\) such that for any \(R>0\) we have \[ \nu\left({z\in \mathrm{Mat}(\mathbb{N},\mathbb{C}):\max_{i,j m} |z_i,j|<R}\right)<{+\infty}. \] Theorem 1.1. If a \(\mathrm{U}(\infty)\times \mathrm{U}(\infty)\)-invariant Borel measure from the class \(\mathfrak{F}(m; \mathrm{Mat}(\mathbb{N},\mathbb{C}))\) is ergodic then it is finite.A similar result holds for infinite Hermitian matrices \(H\subset \mathrm{Mat}(\mathbb{N},\mathbb{C})\). The group \(\mathrm{U}(\infty)\) acts on \(H\) by conjugation. For \(m\in \mathbb{N}\) denote as \(\mathfrak{F}(m,H)\) the space of Borel measures \(\nu\) on \(H\) such that for any \(R>0\) we have \[ \nu\left({h\in H:\max_{i m,j m} |h_{ij}| R}\right)<\infty. \] The main result of the paper is the following.Theorem 1.3. If a \(\mathrm{U}(\infty)\)-invariant measure from the class \(\mathfrak{F}(m,H)\) is ergodic, then it is finite. Reviewer: Victor Sharapov (Dayton) Cited in 3 Documents MSC: 37A15 General groups of measure-preserving transformations and dynamical systems 28D15 General groups of measure-preserving transformations Keywords:infinite-dimensional Lie groups; classification of ergodic measures; orbital measures; weak compactness × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] Bogachev, Vladimir I., Measure theory, II (2007) · Zbl 1120.28001 [2] Borodin, Alexei; Olshanski, Grigori, Infinite random matrices and ergodic measures, Comm. Math. Phys., 223, 1, 87-123 (2001) · Zbl 0987.60020 [3] Bufetov, Alexander I., Ergodic decomposition for measures quasi-invariant under Borel actions of inductively compact groups, Sbornik Mathematics, 205, 2, 39-71 (2014) [4] Olshanski, Grigori, Unitary representations of infinite-dimensional classical groups (Russian) · Zbl 1036.43002 [5] Olshanski, Grigori; Vershik, A. M.; Zhelobenko, D. P., Representation of Lie Groups and Related Topics, 7, 165-189 (1990) · Zbl 0724.22020 [6] Olshanski, Grigori; Vershik, Anatoli, Contemporary mathematical physics, 175, 137-175 (1999) · Zbl 0853.22016 [7] Pickrell, Doug, Measures on infinite-dimensional Grassmann manifolds, J. Funct. Anal., 70, 323-356 (1987) · Zbl 0621.28008 [8] Pickrell, Doug, Mackey analysis of infinite classical motion groups, Pacific J. Math., 150, 1, 139-166 (1991) · Zbl 0739.22016 [9] Rabaoui, Marouane, A Bochner type theorem for inductive limits of Gelfand pairs, Ann. Inst. Fourier (Grenoble), 58, 5, 1551-1573 (2008) · Zbl 1154.22015 [10] Rabaoui, Marouane, Asymptotic harmonic analysis on the space of square complex matrices, J. Lie Theory, 18, 3, 645-670 (2008) · Zbl 1167.22008 [11] Vershik, Anatoly M., A description of invariant measures for actions of certain infinite-dimensional groups, (Russian) Dokl. Akad. Nauk SSSR, 218, 749-752 (1974) · Zbl 0324.28014 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.