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 L1(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.