The order topology on duals of C$^\ast$-algebras and von Neumann algebras

Abstract

For a von Neumann algebra $\mathcal M$ we study the order topology associated with 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 to this, it is proved that the condition of having the order topology associated to the interval $[0,\varphi]$ equal to that 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 linear 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 nonequivalent for all C$^\ast$-algebras that are not of Type $I$.