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

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 )) ((1le p) 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 (1le q, 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 (1le 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 )) 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.