States on 𝑊*-algebras and orthogonal vector measures

Abstract

We show that every state on a W ∗ {W^ * } -algebra A \mathcal {A} without type I 2 {I_2} direct summand is induced by an orthogonal vector measure on A \mathcal {A} . This result may find an application in quantum stochastics [ 1 , 7 ] [1,7] . Particularly, it allows us to find a simple formula for the transition probability between two states on A \mathcal {A} [ 3 , 8 ] [3,8] .