Characterizations of conditional expectations for \(L_ 1(X)\)-valued functions (Q1083750)

In this paper the author considers the problem of characterizing conditional expectations of \(Y=L_ 1(\Omega,{\mathcal A},\mu,E)\) functions, where Y denotes all Bochner integrable functions on the probability space (\(\Omega\),\({\mathcal A},\mu)\) with values in a Banach space E. Taking E as \(L_ 1(X,S,\lambda)\) for some measure space (X,S,\(\lambda)\) with ergodic transformations, the author shows that every constant preserving contractive projection on Y is a conditional expectation operator, when it commutes with these ergodic transformations. The author also proves a few other related results. In this context, the reader should also consult the paper by \textit{D. Landers} and \textit{L. Rogge} [Proc. Am. Math. Soc. 81, 107-110 (1981; Zbl 0471.60013)].