The 'transition probability' in the state space of a * algebra

Let ω, rho be two states of a * -algebra and let us consider representations of this algebra R for which ω and rho are realised as vector states by vectors x and y. The transition probability P(ω, rho) is the supremum of all the numbers (x,y)2, where we run through all such realisations. We derive properties of this straightforward generalisation of the quantum mechanical transition probability and give, in some important cases, an explicit expression of this quantity. (author)