Ergodic unitarily invariant measures on the space of infinite Hermitian matrices
Abstract
Let $H$ be the space of all Hermitian matrices of infinite order and $U(\infty)$ be the inductive limit of the chain $U(1)\subset U(2)\subset...$ 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 Vershik's note \cite{V}. The problem is reduced to studying the limit behavior of orbital integrals of the form $$\int_{B\in\Omega_n}e^{i\op{tr}(AB)}M_n(dB),$$ 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 \cite{V}.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 January 1996
 arXiv:
 arXiv:math/9601215
 Bibcode:
 1996math......1215O
 Keywords:

 Mathematics  Representation Theory;
 Mathematics  Classical Analysis and ODEs;
 Mathematics  Probability
 EPrint:
 In: Contemporary Mathematical Physics. F. A. Berezin's memorial volume. Amer. Math. Transl. Ser. 2, vol. 175 (R. L. Dobrushin et al., eds), 1996, pp. 137175