Let G be a Polish locally compact group acting on a Polish space $${{X}}$$ X with a G-invariant probability measure $$\mu $$ μ . We factorize the integral with respect to $$\mu $$ μ in terms of the integrals with respect to the ergodic measures on X, and show that $$\mathrm {L}^{p}({{X}},\mu )$$ L p ( X , μ ) ( $$1\le p<\infty $$ 1 ≤ p < ∞ ) is G-equivariantly isometrically lattice isomorphic to an $${\mathrm {L}^p}$$ L p -direct integral of the spaces $$\mathrm {L}^{p}({{X}},\lambda )$$ L p ( X , λ ) , where $$\lambda $$ λ ranges over the ergodic measures on X. This yields a disintegration of the canonical representation of G as isometric lattice automorphisms of $$\mathrm {L}^{p}({{X}},\mu )$$ L p ( X , μ ) as an $${\mathrm {L}^p}$$ L p -direct integral of order indecomposable representations. If $$({{X}}^\prime ,\mu ^\prime )$$ ( X ′ , μ ′ ) is a probability space, and, for some $$1\le q<\infty $$ 1 ≤ q < ∞ , G acts in a strongly continuous manner on $$\mathrm {L}^{q}({{X}}^\prime ,\mu ^\prime )$$ L q ( X ′ , μ ′ ) as isometric lattice automorphisms that leave the constants fixed, then G acts on $$\mathrm {L}^{p}({{X}}^{\prime },\mu ^{\prime })$$ L p ( X ′ , μ ′ ) in a similar fashion for all $$1\le p<\infty $$ 1 ≤ p < ∞ . Moreover, there exists an alternative model in which these representations originate from a continuous action of G on a compact Hausdorff space. If $$({{X}}^\prime ,\mu ^\prime )$$ ( X ′ , μ ′ ) is separable, the representation of G on $$\mathrm {L}^p(X^\prime ,\mu ^\prime )$$ L p ( X ′ , μ ′ ) can then be disintegrated into order indecomposable representations. The notions of $${\mathrm {L}^p}$$ L p -direct integrals of Banach spaces and representations that are developed extend those in the literature.
Positivity – Springer Journals
Published: May 6, 2017
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