Noncommutative Yosida-Hewitt theorem in noncommutative Calderón-Lozanovskiĭ spaces
The Yosida-Hewitt theorem asserts that each bounded finitely additive measure \(\mu\) admits a unique decomposition \(\mu=\mu_1+\mu_2\), where \(\mu_1\) is a countably additive measure and \(\mu_2\) is a purely finitely additive measure. As a consequence, any continuous linear functional on \(L^\infty\) may be decomposed into an integral functional generated by an \(L^1\) function and a functional that is disjoint to any integral functional. The authors provide a Yosida-Hewitt type decomposition for continuous linear functionals on noncommutative Calderón-{L}ozanovskiĭ spaces. Namely: Theorem. Let \(\mathcal{M}\) be a diffuse semifinite von Neumann algebra (or finite von Neumann algebra) and \(\varphi\) be an Orlicz function. Assume that \(E\) is an \(\alpha\)-convex symmetric quasi-Banach space on \(I\) for some \(0<\alpha<\infty\). Then every \(\ell\in E_{\varphi}(\mathcal{M})^*\) admits a unique decomposition \[ \ell=\ell_n+\ell_s, \] where \(\ell_s\in E_{\varphi}(\mathcal{M})_s^*\) and \(\ell_n\in E_{\varphi}(\mathcal{M})_n^*\) is given by \[ \ell_n(x)=\tau(xy)\quad \left(x\in E_{\varphi}(\mathcal{M})\right) \] for a unique \(y\in E_{\varphi}(\mathcal{M})^\times\). Furthermore, \begin{itemize} \item[(i)] if \(\ell\) is Hermitian, then \(y=y^*\); \item[(ii)] if \(\ell\geq0\), then \(y\geq0\) and \(\ell_s\geq0\); \item[(iii)] if \(E_{\varphi}(\mathcal{M})^\times=0\), then \(\ell_n=0\). \end{itemize}
noncommutative Calderón-Lozanovskiĭ spaces
scientific article; zbMATH DE number 7222736
scientific article