Almost Sure

4 January 20

Operator Topologies

We previously defined the notion of positive linear maps and states on *-algebras, and noted that there always exists seminorms defining the {L^2} and {L^\infty} 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 {(\mathcal A,p)} is straightforward to define. A net {a_\alpha\in\mathcal A} tends weakly to the limit {a} if and only if {p(xa_\alpha y)\rightarrow p(xay)} for all {x,y\in\mathcal A}. (more…)

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