Module homomorphisms of a von Neumann algebra into its center

Herbert Halpern
1969 Transactions of the American Mathematical Society  
1. Introduction. A tool for studying von Neumann algebras is the space of CT-weakly continuous linear functionals. There are certain limitations in the applicability of this tool because the center of the algebra is somehow submerged and on the whole it seems a-weakly continuous functionals are more suitable for studying factor algebras. In this article we propose another tool, viz. the a-weakly continuous module homomorphisms of the von Neumann algebra si, considered as a module over its
more » ... S, into 2f. First it is shown that every a-weakly continuous linear functional can be written as the composition of a a-weakly continuous module homomorphism into 3f and a a-weakly continuous linear functional on 3f. For these module homomorphisms there is obtained a specific form which generalizes the well-known form of a-weakly continuous linear functionals ([1], [16] ). Certain facts concerning decomposition theory result. Further si is the dual (in terms of module homomorphisms) of the space of all a-weakly continuous module homomorphisms of si into 2£. As an example of the applicability of this tool a type I algebra is described as the second dual (in terms of module homomorphisms) of a module. Further applications are contained in a later paper. It seems reasonable to conjecture from the ensuing proofs that considering the module structure might further illuminate the relation between a von Neumann
doi:10.1090/s0002-9947-1969-0241986-5 fatcat:ftdrlpsyivgnhpvhv5jyhikcgq