Characterization of conditional expectations for $M$-space-valued functions

Ryohei Miyadera
1993
Introduction Let (Ω, Jl, μ) be a probability space, E a Banach space. We consider constant-preserving contractive projections of L^Ω, <Jl, μ, E) into itself. If E=R or E is a strictly-convex Banach space, then it is known (Ando [2], Douglas [3] and Landers and Rogge [6] ) that such operators coincide precisely with the conditional expectation operators. If E=L 1 (X, 5, λ, R), where (X, S, λ) is a localizable measure space, then the author [8] proved that such operators which are translation
more » ... are translation invariant coincide with the conditional expectation operators. If E=L 00 (X, 5, λ, R), where (X, S y λ) is a measure space, and the dimension of E is bigger than 2, then author [9] proved that such operators coincide with the conditional expectation operators. On the other hand if E= Loo(X, S, λ, R) and the dimension of E is 2, then the author [9] proved that such operators can be expressed as a linear combination of two conditional expectation operators. In this paper we deal with the case that E is an M-space. An Loo-space is an M-space, and hence this paper contains the result of the author [9] as a special case. If E is an M-space, whose dimension is bigger than 2, then such operators coincide with conditional expectation operators. If E is an M-space with unit, i.e., the unit ball in E has a least upper bound, then we can prove many of lemmas in this paper by easier way. In this paper we do not assume that E is an M-space with unit.
doi:10.18910/9980 fatcat:swvxecdsbbd43f55igzpiytvba