  It is easy to see that every character (i.e. unital *-homomorphism to the
complex numbers) of a commutative unital associative *-algebra is a pure state
(i.e. extreme point in the convex set of all normalized positive linear
functionals). This article gives sufficient conditions for the converse to be
true as well. In order to formulate these results together with similar ones,
e.g. for locally convex *-algebras, the notion of an abstract O*-algebra
(unital associative *-algebra with an order defined by positive linear
functionals) is introduced. Many concepts and intermediary results discussed
here also apply to the non-commutative case.
