Disintegration of positive isometric group representations on ${\mathrm{L}}^p$-spaces

Abstract

Let $G$ be a Polish locally compact group acting on a Polish space $X$ with a $G$-invariant probability measure $\mu$. We factorize the integral with respect to $\mu$ in terms of the integrals with respect to the ergodic measures on $X$, and show that ${\mathrm{L}}^p(X,\mu )$ ($1\le p<\infty$) is $G$-equivariantly isometrically lattice isomorphic to an ${\mathrm{L}}^p$-direct integral of the spaces ${\mathrm{L}}^p(X,\lambda )$, where $\lambda$ ranges over the ergodic measures on $X$. This yields a disintegration of the canonical representation of $G$ as isometric lattice automorphisms of ${\mathrm{L}}^p(X,\mu )$ as an ${\mathrm{L}}^p$-direct integral of order indecomposable representations. If $(X^{\prime },{\mu }^{\prime })$ is a probability space, and, for some $1\le q<\infty$, $G$ acts in a strongly continuous manner on ${\mathrm{L}}^q(X^{\prime },{\mu }^{\prime })$ as isometric lattice automorphisms that leave the constants fixed, then $G$ acts on ${\mathrm{L}}^p(X^{\prime },{\mu }^{\prime })$ in a similar fashion for all $1\le p<\infty$. Moreover, there exists an alternative model in which these representations originate from a continuous action of $G$ on a compact Hausdorff space. If $(X^{\prime },{\mu }^{\prime })$ is separable, the representation of $G$ on ${\mathrm{L}}^p(X^{\prime },{\mu }^{\prime })$ 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.

Section on future perspectives added. 35 pages. To appear in Positivity

References in corpus (1)

• Positive representations of $C_0(X)$. I

Cited by in corpus (3)

• Sobolev space of functions valued in a monotone Banach family
• Ergodic Decomposition of Dirichlet Forms via Direct Integrals and Applications
• Decomposable operators acting between distinct $L^p$-direct integrals of Banach spaces