The “transition probability” in the state space of a 1-algebra

Let ω,? be two states of a 1-algebra and let us consider representations of this algebra R for which ω and ? are realized as vector states by vectors x and y. The transition probability P(ω,?) 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.