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 ${\mathcal{M}}_{\ast }^s$ and to intervals of the predual ${\mathcal{M}}_{\ast }$. It is shown that the order topology on ${\mathcal{M}}_{\ast }^s$ 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 $[0,\phi ]$ equal to that induced by the norm for every $\phi \in {\mathcal{M}}_{\ast }^{+}$, is necessary and sufficient for the commutativity of $\mathcal{M}$. It is also proved that if $\phi$ is a positive bounded linear functional on a C${}^{\ast }$-algebra $\mathcal{A}$, then the norm-null sequences in $[0,\phi ]$ coincide with the null sequences with respect to the order topology on $[0,\phi ]$ if and only if the von Neumann algebra ${\pi }_{\phi }(\mathcal{A}{)}^{\prime }$ is of finite type (where ${\pi }_{\phi }$ 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 $I$.

12 pages