The order topology for a von Neumann algebra (Q2787144)

The authors study the relationship of order topologies and the Mackey topology on certain subsets of von Neumann algebras.NEWLINENEWLINEIn a partially ordered set \((P,\leq)\), a net \((x_\gamma)\) is said to order converge to \(x\in P\) if there exist nets \((y_\gamma)\), \((z_\gamma)\) in \(P\) such that \(y_\gamma\leq x_\gamma\leq z_\gamma\) and \(\bigvee y_\gamma=x=\bigwedge z_\gamma\). By definition, the order topology \(\tau_o(P)\) of \(P\) is the finest topology preserving order convergence. The sequential counterpart of the sequential order topology of \(P\), denoted by \(\tau_{os}(P)\), satisfies (trivially) that \(\tau_{o}(P)\subset\tau_{so}(P)\). In general, this inclusion is strict; both topologies are \(T_1\) but not necessarily Hausdorff on ordered vector spaces, hence not necessarily linear.NEWLINENEWLINERecall that a topology is called sequential if sequentially closed sets are closed.NEWLINENEWLINELet \(M\) be a von Neumann algebra with predual \(M_*\), \(M_{sa}\) its selfadjoint part, \(M_{sa}^1\) the unit ball of the latter and \(P(M)\) the complete lattice of projections of \(M\). The order considered on these three parts \(M_{sa}\), \(M_{sa}^1\), \(P(M)\) is the usual operator order. Finally, the usual locally convex topologies on \(M\) are considered: the Mackey topology \(\tau=\tau(M,M_*)\), the \(\sigma\)-strong and \(\sigma\)-strong\(^*\) topologies \(\sigma=\sigma(M,M_*)\) and \(\sigma^*=\sigma^*(M,M_*)\), the weak and strong operator topologies \(\tau_w\) and \(\tau_s\).NEWLINENEWLINEIn the paper under review, the relationship of the order topologies and the Mackey topology (and other locally convex topologies) is studied systematically for \(M_{sa}\), \(M_{sa}^1\) and \(P(M)\). Despite the different nature of order and locally convex topologies, there are some quite satisfactory results. Let us mention four main results. {\parindent=0.6cm \begin{itemize}\item[--] \(M\) is abelian if and only if \(\tau_o(M_{sa})\) and \(\tau_o(M_{sa}^1)\) coincide on \(P(M)\), if and only if \(\tau_o(M_{sa}^1)\) equals the restriction of \(\tau_o(M_{sa})\) to \(M_{sa}^1\). \item[--] \(M\) is \(\sigma\)-finite if and only if \(\tau_o\) and \(\tau_{os}\) coincide on \(P(M)\), if and only if they coincide on \(M_{sa}\), if and only if they coincide on \(M_{sa}^1\). \item[--] \(M\) is \(\sigma\)-finite and \(\tau_o(M_{sa})\) is a linear topology if and only if the restriction of the Mackey topology to \(M_{sa}\) equals \(\tau_{os}(M_{sa})\), if and only if the Mackey topology is sequential. \item[--] If \(M\) is \(\sigma\)-finite, then \(M\) is of finite type if and only if the restriction of \(\tau_o(M_{sa}^1)\) to \(P(M)\) equals \(\tau_o(P(M))\), if and only if \(\sigma\)-strong null sequences in \(M_{sa}^1\) are \(\tau_{os}(M_{sa}^1)\) null. NEWLINENEWLINE\end{itemize}} An important tool is the notion of the mixed topology introduced in [\textit{M. Nowak}, Pac. J. Math. 140, No. 1, 155--161 (1989; Zbl 0696.46028)]; it is shown that the mixed topology of the uniform and the \(\sigma\)-strong\(^*\) topology coincides with the Mackey topology on \(M\).