An Elementary Proof of Douglas′ Theorem on Contractive Projections on L1-Spaces

Douglas ( Pacific J. Math. 15 (1965), 443-462) has shown that the conditional expectation operators are the only contractive projections on L 1(σ) that leave the constant functions invariant. This remarkable result has applications to several areas and our objective is to present here an elementary and self-contained proof. Douglas* theorem has been proven and generalized in various contexts by many authors; for details see Ando ( Pacific J. Math. 17 (1966), 391-405), Bernau and Lacey ( Pacific J. Math. 53 (1974), 21-41), and Dodds et al. ( Pacific J. Math. 141 (1990), 55-77) and the references therein.