$\mathrm{L}^p$ -空间上正等距群表示的分解
Disintegration of positive isometric group representations on $\mathrm{L}^p$-spaces
摘要：
设$G$是作用在具有$G$ -不变概率测度$μ$的波兰空间$X$上的波兰局部紧群.我们因式分解的积分相对于$μ$的积分关于$X$上的遍历措施，并表明$\mathrm{L}^p(X,μ)$（$1\leq p<\infty$）是$G$ -等变等距格同构的$\mathrm{L}^p$ -直接积分的空间$\mathrm{L}^{p}(X,λ)$，其中$λ$范围在$X$上的遍历措施。这产生了一个分解的正则表示的$G$作为等距格自同构的$\mathrm{L}^p(X,μ)$作为一个$\mathrm{L}^p$ -直接积分的顺序不可分解的表示。如果$(X^\prime,μ^\prime)$是一个概率空间，并且对于某些$1\leq q<\infty$，$G$以强连续的方式作用在$\mathrm{L}^q(X^\prime,μ^\prime)$上，作为保持常数不变的等距格自同构，则$G$对所有$1\leq p<\infty$以类似的方式作用在$\mathrm{L}^{p}(X^{\prime},μ^{\prime})$上。此外，存在一种替代模型，其中这些表示源自$G$在紧致豪斯多夫空间上的连续作用。如果$(X^\prime,μ^\prime)$是可分的，那么$G$在$\mathrm{L}^p(X^\prime,μ^\prime)$上的表示就可以分解为顺序不可分解的表示。$\mathrm{L}^p$的概念-直接积分的Banach空间和表示，扩展了那些在文献中。
英文摘要：
Let $G$ be a Polish locally compact group acting on a Polish space $X$ with a $G$-invariant probability measure $μ$. We factorize the integral with respect to $μ$ in terms of the integrals with respect to the ergodic measures on $X$, and show that $\mathrm{L}^p(X,μ)$ ($1\leq p<\infty$) is $G$-equivariantly isometrically lattice isomorphic to an $\mathrm{L}^p$-direct integral of the spaces $\mathrm{L}^{p}(X,λ)$, where $λ$ ranges over the ergodic measures on $X$. This yields a disintegration of the canonical representation of $G$ as isometric lattice automorphisms of $\mathrm{L}^p(X,μ)$ as an $\mathrm{L}^p$-direct integral of order indecomposable representations. If $(X^\prime,μ^\prime)$ is a probability space, and, for some $1\leq q<\infty$, $G$ acts in a strongly continuous manner on $\mathrm{L}^q(X^\prime,μ^\prime)$ as isometric lattice automorphisms that leave the constants fixed, then $G$ acts on $\mathrm{L}^{p}(X^{\prime},μ^{\prime})$ in a similar fashion for all $1\leq p<\infty$. Moreover, there exists an alternative model in which these representations originate from a continuous action of $G$ on a compact Hausdorff space. If $(X^\prime,μ^\prime)$ is separable, the representation of $G$ on $\mathrm{L}^p(X^\prime,μ^\prime)$ can then be disintegrated into order indecomposable representations. The notions of $\mathrm{L}^p$-direct integrals of Banach spaces and representations that are developed extend those in the literature.
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