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