Why be normal?

Suppose we have an algebra AA of quantum observables. What virtues must a function ω:A→Cω:A→C exhibit in order to qualify as a quantum state? One virtue familiar from density operator states is countable additivity  : a density operator ρρ on HH determines a countably additive probability distribution over H'sH's closed subspaces; such a probability distribution corresponds to what's known as a normal state on the von Neumann algebra B(H)B(H) of bounded operators on the Hilbert space. This essay investigates the virtue of normality, by presenting a number of physical situations and/or interpretive impulses that might tempt one to acknowledge states that are not normal, and by adducing reasons to resist these temptations.