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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

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.

MSC:

37A15 General groups of measure-preserving transformations and dynamical systems
28D15 General groups of measure-preserving transformations

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