The order topology on duals of C*-algebras and von Neumann algebras

摘要

For a von Neumann algebra M, we study the order topology associated to the hermitian part M-*(s), and to intervals of the predual M-*. It is shown that the order topology on 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, phi], equal to the topology induced by the norm, for every phi is an element of M-*(+), is necessary and sufficient for the commutativity of M. It is also proved that if phi is a positive bounded functional on a C*-algebra 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)(A)' 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 inequivalent for all C*-algebras that are not of type I.