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Ergodic unitarily invariant measures on the space of infinite Hermitian matrices. (English) Zbl 0853.22016
Dobrushin, R. L. (ed.) et al., Contemporary mathematical physics. F. A. Berezin memorial volume. Transl. ed. by A. B. Sossinsky. Providence, RI: American Mathematical Society. Transl., Ser. 2, Am. Math. Soc. 175(31), 137-175 (1996).
Summary: Let \(H\) be the space of all Hermitian matrices of infinite order and \(U (\infty)\) be the inductive limit of the chain \(U(1) \subset U(2) \subset \cdots\) of compact unitary groups. The group \(U (\infty)\) operates on the space \(H\) by conjugations, and our aim is to classify the ergodic \(U (\infty)\)-invariant probability measures on \(H\) by making use of a general asymptotic approach proposed in A. M. Vershik’s note [Sov. Math., Dokl. 15, 1396-1400 (1974); translation from Dokl. Akad. Nauk SSSR 218, 749-752 (1974; Zbl 0324.28014)]. The problem is reduced to studying the limit behavior of orbital integrals of the form \[ \int_{B \in \Omega_n} e^{i \text{tr} (AB)} M_n (dB), \] where \(A\) is a fixed \(\infty \times \infty\) Hermitian matrix with finitely many nonzero entries, \(\Omega_n\) is a \(U(n)\)-orbit in the space of \(n \times n\) Hermitian matrices, \(M_n\) is the normalized \(U(n)\)-invariant measure on the orbit \(\Omega_n\), and \(n \to \infty\).
We also present a detailed proof of an ergodic theorem for inductive limits of compact groups that has been announced in [Vershik, op. cit.]. There is a remarkable link between our subject and I. J. Schoenberg’s [J. Analyse Math. 1, 331-374 (1951; Zbl 0045.37602)] theory of totally positive functions, and our approach leads to a new proof of Schoenberg’s [op. cit.] main theorem, originally proved by function-theoretic methods. On the other hand, our results have a representation-theoretic interpretation, because the ergodic \(U (\infty)\)-invariant measures on \(H\) determine irreducible unitary spherical representations of an infinite-dimensional Cartan motion group. The present paper is closely connected with a series of articles by S. V. Kerov and the authors on the asymptotic representation theory of “big” groups, but it can be read independently.
For the entire collection see [Zbl 0845.00041].

22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties
28C10 Set functions and measures on topological groups or semigroups, Haar measures, invariant measures