The von Neumann algebra characterization theorems
There are two standard characterization theorems that give necessary and sufficient conditions for a \(C^*\)-algebra to be isomorphic to a von Neumann algebra. Sakai's states that these \(C^*\)-algebras are precisely those that are (isometric to) the norm-dual space of some Banach space. The author's states that these \(C^*\)-algebras are the ones that are monotone closed and possess a separating family of normal states. In this paper, the author derives the first characterization from the second. Complete, streamlined proofs are given.
necessary and sufficient conditions for a \(C^*\)-algebra to be
necessary and sufficient conditions for a \(C^*\)- algebra to be isomorphic to a von Neumann algebra
scientific article; zbMATH DE number 3921283