In quantum mechanics, a qausi-state on an algebra of observables is a function that is required to satisfy the axioms of a genuine state (linearity and positivity) only on the poset of commutative subalgebras of .
While therefore the condition on quasi-states is much weaker than that for states, Gleason's theorem asserts that if for , then all quasi-states are in fact states.
Notice that a quasi-state is naturally regarded as an actual state, but internal to the ringed topos over the poset of commutative subalgebras of . Therefore Gleason’s theorem is one of the motivations for regarding this ringed topos as the quantum phase space (“Bohrification”.) The other is the Kochen-Specker theorem.