Suppose that N is a von Neumann algebra and that φ is a normal semifinite trace on N. We study bands, projection bands, and band projections in the setting of non- commutative Banach function spaces. Let S(N, φ) be the set of all φ-measurable operators. A subspace M of an N-bimodule X ⊆ S(N, φ) is called a projection band if X = M ⊕ M⊥ , where M⊥ = {x ∈ (X , φ) : x y = x ∗ y = 0 for all y ∈ M}. We show that any band in a noncommutative Banach function space E(φ) satisfying the Beppo–Levi property with an order continuous norm is a projection band. More- over, we characterize bands in E(φ) associated to a σ -φ-finite von Neumann algebra. A linear map on E(φ) is said to be projection-preserving if for every projection p ∈ E(φ), ( p) is a projection. Among other results, we show that if is a normal projection-preserving linear map such that (x) ≤ x (x ≥ 0) and p is a σ -φ-finite projection, then ( px p) = q xq (x ∈ E(φ)) for some projection q ≤ p. Furthermore, there exist two completely additive semifinite traces φ1 , φ2 ≤ φ on N with φ1 +φ2 = φ such that for every σ -φ-finite projection p, there are two orthogonal projections p1 and p2 with p = p1 + p2 satisfying φ1 ( p2 ) = φ2 ( p1 ) = 0.

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