Suppose (X,Σ,μ) is a
measure space, 1 ≦ p < ∞ and P≠ 2. Let Lp= Lp(X,Σ,μ) be the usual space of
equivalence classes of Σ-measurable functions f such that |f|p is integrable. A
contractive projection on Lp is a linear operator P : Lp→ Lp such that P2= P and
∥P∥≦ 1. In this paper we give a complete description of such contractive projections
in terms of conditional expectation operators. We also show that a closed subspace
M of Lp is the range of a contractive projection if and only if M is isometrically
isomorphic to another Lp-space. Another sufficient condition shows, in particular,
that every closed vector sublattice of an Lp-space is the range of a positive
contractive projection.
