A Topology for Operator Modules overW*-Algebras

Given a von Neumann algebraRon a Hilbert space H, the so-calledR-topology is introduced into B(H), which is weaker than the norm and stronger than the ultrastrong operator topology. A rightR-submoduleXof B(H) is closed in theR-topology if and only if for eachb∈B(H) the right ideal, consisting of alla∈Rsuch thatba∈X, is weak* closed inR. Equivalently,Xis closed in theR-topology if and only if for eachb∈B(H) and each orthogonal family of projectionseiinRwith the sum 1 the conditionbei∈Xfor alliimplies thatb∈X.