A characterization of complex \(L_1\)-preduals via a complex barycentric mapping. (Q2798067)

Let \(\mathbb F\) denote the field \(\mathbb R\) of real numbers or the field \(\mathbb C\) of complex numbers and let \(K\) be a compact space. Let \({\mathcal M}(K,{\mathbb F})\) denote the Banach space of all \(\mathbb F\)-valued Radon measures on \(K\), provided with the weak\(^*\) topology coming from the natural duality with \({\mathcal C}(K,{\mathbb F})\). Consider an assignment \(K\ni k\longmapsto \rho_k\in {\mathcal M}(K,\mathbb F)\) and for \(f\in{\mathcal C}(K,{\mathbb F})\) put \(f_\rho(k):= \int_K f\text{d}\rho_k,\;k\in K\). Define \(A_\rho:= \big\{f\in {\mathcal C}(K,{\mathbb F}):\;f=f_\rho\}\). Then \(\rho\) is called an \(\mathbb F\)-barycentric mapping if \(\|\rho_k\|\leq1\) for every \(k\in K\), if the function \(f_\rho\) is universally measurable for every \(f\in{\mathcal C}(K,{\mathbb F})\), and if every \(\nu\in{\mathcal M}(K,\mathbb F)\) which annihilates on \(A_\rho\) is already \(0\).NEWLINENEWLINE The paper deals with the following Theorem. (a) If \(\rho\) is \(\mathbb F\)-barycentric, then the Banach space \(A_\rho\) is an \(\mathbb F\)-\(L_1\)-predual, i.e., the dual space \(A_\rho{}^*\) is isometric to \(\mathbb F\)-\(L_1(\Omega,{\mathcal S},\mu)\) where \((\Omega,{\mathcal S},\mu)\) is a suitable measure space. (b) Reversely, if \(X\) is an \(\mathbb F\)-\(L_1\)-predual, then there exist a compact space \(K\) and an \(\mathbb F\)-barycentric mapping \(\rho:K\longrightarrow {\mathcal M}(K,\mathbb F)\) such that \(X\) is isometric to \(A_\rho\). The real case, that is \(\mathbb F:=\mathbb R\), was proved by \textit{J. B. Bednar} and \textit{H. E. Lacey} [Pac. J. Math. 41, 13--24 (1972; Zbl 0214.37502)]. The paper under review deals with the complex case, i.e., \(\mathbb F:=\mathbb C\). While the proof of (a) is rather simple and similar to the real case, the proof of (b) is longer and is not an imitation of the real case.