  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 $I$.
