Transition Probabilities of Positive Linear Functionals on $*$-Algebras

Abstract

Using unbounded Hilbert space representations basic results on the transition probability of positive linear functionals $f$ and $g$ on a unital *-algebra are obtained. The main assumption is the essential self-adjointness of GNS representations $π_f$ and $π_g$. Applications to functionals given by density matrices and by integrals and to vector functionals on the Weyl algebra are given.