Operators preserving disjointness on rearrangement invariant spaces

Let X and Y be two rearrangement invariant spaces on a measure space (, , ) with a finite, nonatomic measure . We show that if there exists a non-zero order continuous disjointness preserving operator T: X - Y, then X C.Y. This result has many consequences. For example, if T: L P (, , ) -> L q (, , ) (0 < p < q < oo) preserves disjointness, then T = 0.

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