(i) Let be a state on a -algebra . Suppose that for all unitary elements u∈A. Show that φ is a pure state. [Hint: ]
(ii) Let be a multiplicative functional on a -algebra . Show that is a pure state on .
(iii) Show that the pure and multiplicative states coincide for commutative .
I managed to work out the first two problems but I have no idea about the last one. How to see from being an extreme element in the state space of a commutative that the extreme element is multiplicative?