A characterization of conditional expectations for \(L_{\infty}(X)\)- valued functions (Q1103936)

Let \(E=L_{\infty}(X,S,\lambda,R)\), where (X,S,\(\lambda)\) is a measure space and \(Y=L_ 1(\Omega,{\mathcal A},\mu,E)\) the space of E-valued Bochner integrable functions on a probability space (\(\Omega\),\({\mathcal A},\mu)\). We consider a constant-preserving contractive projection Q of Y into itself. If there exist pairwise disjoint sets \(X_ 1\), \(X_ 2\) and \(X_ 3\) of positive measure such that \(X=X_ 1\cup X_ 2\cup X_ 3\), then \(Q(f)=f^{{\mathcal B}}\) for some \(\sigma\)-subalgebra \({\mathcal B}\) of \({\mathcal A}\), where \(f^{{\mathcal B}}\) is the conditional expectation of f given \({\mathcal B}.\) On the other hand if \(E\cong R^ 2\) with the norm \(\| (x,y)\| =| x| \vee | y|\), then \[ Q((f,g)=(1/2(f^{{\mathcal B}}+g^{{\mathcal B}}+f^{{\mathcal C}}-g^{{\mathcal C}}),\quad 1/2(f^{{\mathcal B}}+g^{{\mathcal B}}+g^{{\mathcal C}}-f^{c})) \] for each \((f,g)\in L_ 1(\Omega,{\mathcal A},\mu,R^ 2)\) for some \(\sigma\)-subalgebras \({\mathcal B}\) and \({\mathcal C}\) of \({\mathcal A}\). After the publication of this paper the author proved a similar result for the case when E is the space of bounded continuous functions on a normal topological space. Generalizations of these results to the case when (\(\Omega\),\({\mathcal A},\mu)\) is a general measure space are also achieved by the author.