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DISINTEGRATING POSITIVE GROUP REPRESENTATIONS ON L p -SPACES INTO ORDER INDECOMPOSABLES

Let G be a Polish locally compact group acting on a Polish space X with a G-invariant probability measure . We disintegrate the associated positive representation of G on L p (X; ) as a p-direct integral of order inde- composable representations, for all p 2 (1;1). This disintegration is for the most part obtained by showing that L p (X; ) is isometrically lattice isomorphic to the p-direct integral of L p (X; ), with ranging over the ergodic measures on X. The pertinent notion of a direct integral of Banach spaces and Banach lattices extends that in the literature. As a corollary of our main result we disintegrate an action of a compact Polish group as isometric Markov operators on L 1 (X; ), for (X; ) a prob- ability space such that L 1 (X; ) is separable, as a 1-direct integral of order indecomposable representations.