The order topology on duals of $C^\ast $-algebras and von Neumann algebras
Volume 254 / 2020
Abstract
For a von Neumann algebra $\mathcal M$, we study the order topology associated to the hermitian part $\mathcal M_*^s$, and to intervals of the predual $\mathcal M_*$. It is shown that the order topology on $\mathcal M_*^s$ coincides with the topology induced by the norm. In contrast, it is proved that the condition of having the order topology, associated to the interval $[0,\varphi ]$, equal to the topology induced by the norm, for every $\varphi \in \mathcal M_*^+$, is necessary and sufficient for the commutativity of $\mathcal M$. It is also proved that if $\varphi $ is a positive bounded functional on a $C^\ast$-algebra $\mathcal A{}$, then the norm-null sequences in $[0,\varphi ]$ coincide with the null sequences, with respect to the order topology on $[0,\varphi ]$, if and only if the von Neumann algebra $\pi _\varphi (\mathcal A)’$ is of finite type (where $\pi _\varphi $ denotes the corresponding GNS representation). This fact allows us to give a new topological characterization of finite von Neumann algebras. Moreover, we demonstrate that convergence to zero for norm and order topology, on order-bounded parts of dual spaces, are inequivalent for all $C^\ast$-algebras that are not of type I.
