Normal Maps

Given two *-probability spaces {(\mathcal A,p)} and {(\mathcal A^\prime,p^\prime)}, we want to consider maps {\varphi\colon\mathcal A\rightarrow\mathcal A^\prime}. For example, we can look homomorphisms, which preserve the *-algebra operations, and can also consider restricting to state-preserving maps satisfying {p^\prime(\varphi(a))=p(a)}. In algebraic probability theory, however, it is often necessary to include a continuity condition, leading to the idea of normal maps, which I look at in this post. In fact, as we will see, all *-homomorphisms between commutative probability spaces which preserve the state are normal, so this concept is most important in the noncommutative setting.

In contrast to the previous few posts on algebraic probability, the current post is a bit of a gear-change. We are still concerned with with the basic concepts of *-algebras and states. However, the main theorem stated below, which reduces to the Radon-Nikodym theorem in the commutative case, is deeper and much more difficult to prove than the relatively simple results with which I have been concerned with so far. Continue reading “Normal Maps”

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}. Continue reading “Operator Topologies”