# 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”