无限埃尔米特矩阵空间上的遍历酉不变测度
Ergodic unitarily invariant measures on the space of infinite Hermitian matrices
摘要：
设$H$是所有无限阶埃尔米特矩阵的空间，$U(\infty)$是紧酉群链$U(1)\subset U(2)\subset...$的归纳极限.群$U(\infty)$通过共轭作用在空间$H$上，我们的目的是利用Vershik的注\cite{V}中提出的一般渐近方法对$H$上的遍历$U(\infty)$不变概率测度进行分类.这个问题归结为研究轨道积分的极限行为的形式$$\int_{B\inΩ_n}e^{i\op{tr}(AB)}M_n(dB),$$其中$A$是一个固定的$\infty\times\infty$厄米矩阵与许多非零元素，$Ω_n$是一个$U(n)$ -轨道在空间中的$n\times n$埃尔米特矩阵，$M_n$是正规化的$U(n)$ -不变的措施轨道$Ω_n$，和$n\to\infty$ .我们还提出了一个详细的证明遍历定理的归纳极限的紧群，已宣布在\cite{V}。
英文摘要：
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 $$\int_{B\inΩ_n}e^{i\op{tr}(AB)}M_n(dB),$$ where $A$ is a fixed $\infty\times\infty$ Hermitian matrix with finitely many nonzero entries, $Ω_n$ 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 $Ω_n$, 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}.
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