A non-commutative Yosida-Hewitt theorem and convex sets of measurable operators closed locally in measure
Let \(M\) be a semifinite von Neumann algebra with a normal faithful semifinite trace \(\tau\). Let \(E(M,\tau)\) be a noncommutative symmetric space associated with \((M,\tau)\) corresponding to a symmetric function space \(E\). Using the noncommutative Köthe duality, the following general form of the Yosida-Hewitt decomposition is proved: Suppose that \(M\subset E(M,\tau)\) or that \(E\) is fully symmetric. If \(\varphi\in E(M,\tau)^\ast\), then there exists a unique decomposition \(\varphi=\tau(a\cdot)+\varphi_s\,,\) where \( a\) is from the Köthe dual of \(E(M,\tau)\) and \(\varphi_s\in E(M,\tau)^\ast\) is singular. (A functional is singular if it vanishes on some order dense ideal.) This permits to define the Yosida-Hewitt projection and characterize, in terms of this projection, bounded convex sets in \(E(M,\tau)\) which are closed locally in measure. Based on this, a Hahn-Banach type result as well as the finite intersection property for convex sets closed locally in measure are established.
scientific article; zbMATH DE number 5070771