The "transition probability" in the state space of a ∗-algebra

Let ω,ϱ be two states of a ∗-algebra and let us consider representations of this algebra R for which ω and ϱ are realized as vector states by vectors x and y. The transition probability P( ω, ϱ) is the spectrum of all the numbers |( x, y)| ² taken over all such realizations. We derive properties of this straightforward generalization of the quantum mechanical transition probability and give, in some important cases, an explicit expression for this quantity.