Operators preserving disjointness on rearrangement invariant spaces

Abstract

Let X and Y be two rearrangement invariant spaces on a measure space (Ω, Σ, μ) with a finite, nonatomic measure μ . 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: LP(Ω, Σ, μ) -> Lq(Ω, Σ, μ) (0 < p < q < oo) preserves disjointness, then T = 0. 1. Notation and preliminary facts. Recall that a (linear) operator T: X —• Y between vector lattices is said to be a disjointness preserving operator if |JCI| Λ \x2 = 0 in X implies \Tx{ Λ \Tx2 = 0 in Y. All vector lattices are assumed to be Archimedean, and all operators on normed or linear metric spaces are assumed to be continuous. Let (Ω, Σ, μ) be a measure space with a finite σ-additive nonatomic measure and S(Ω,Σ,μ) be the space of all (equivalence classes of) measurable real valued functions. Throughout the work we will use the representation of the space S as the space Coo(Q) of all continuous extended functions on the Stone space Q of S. (See [10] for details.) We retain the same notation μ for the corresponding measure on Q, which is defined on the σ-algebra ΣQ consisting of all subsets of the form (E\N) U (N\E), where E is a clopen (closed and open) subset of Q and N is a first category subset of Q. It is well known that μ(D) = 0 if and only if D is a nowhere dense subset of Q. (Any extremally disconnected space Q with such a measure is sometimes called a hyperstonian space.) A subspace X of 5(Ω, Σ, μ) is called a rearrangement invariant (r.i.) ideal if