Topologies on Central Extensions of Von Neumann Algebras

Abstract

Given a von Neumann algebra $M$ we consider the central extension $E(M)$ of $M.$ We introduce the topology $t_c(M)$ on $E(M)$ generated by a center-valued norm and prove that it coincides with the topology of convergence locally in measure on $E(M)$ if and only if $M$ does not have direct summands of type II. We also show that $t_c(M)$ restricted on the set $E(M)_h$ of self-adjoint elements of $E(M)$ coincides with the order topology on $E(M)_h$ if and only if $M$ is a $\sigma$-finite type I$_{fin}$ von Neumann algebra.