We previously defined the notion of positive linear maps and states on *-algebras, and noted that there always exists seminorms defining the and topologies. However, for applications to noncommutative probability theory, these are often not the most convenient modes of convergence to be using. Instead, the *weak*, *strong*, *ultraweak* and *ultrastrong* operator topologies can be used. This, rather technical post, is intended to introduce these concepts and prove their first properties.

Weak convergence on a *-probability space is straightforward to define. A net tends weakly to the limit if and only if for all . (more…)