Definition 1. Let be a C*-algebra and . is positive if it is self-adjoint and . We denote the set of positive elements of by and write for all . We also write if .
It turns out that if is a C*-subalgebra of , then . Hence .
Definition 2. A functional is positive if (hence ). A state is a positive linear functional such that . The set of all states is called the state space and is denoted .
If is a positive linear functional, it defines a pre inner product on :
If is a Hilbert space, we can endow a *-algebra structure on , the space of linear operators on where multiplication is composition, and if , then is defined as the adjoint of .
Definition 3. Let be a C*-algebra and be a Hilbert space. A representation is a *-homomorphism . A subrepresentation is a representation where is a closed subspace of which is invariant under action from . A representation is irreducible if it has no nontrivial subrepresentations. A representation is faithful if .
We now present an important connection discovered by Gelfand, Naimark, and Segal.
Let be a C*-algebra and be a positive linear functional on . Define
is a closed left ideal in , and is an inner product space. Let be its Hilbert completion. Now define by action of : . is clearly a representation, and is called the GNS representation of associated with .
If is unital, let denote the image of in the completion/quotient composition . Then induces a positive linear functional on defined by
Theorem 4. Let be positive linear functionals on a C*-algebra such that (meaning for all ). Then there is a unique operator such that and for all .
This is a generalization of the Radon-Nikodym theorem (which is special case for where is a finite measure space and ).
Definition 5. We define an extreme point in topological vector space as a point that does not belong to any open line segment in . A pure state is an extreme point in . The set of pure states of is denoted .
It follows that .
Proposition 6. Let be a state on a C*-algebra . Then is irreducible if and only if is pure.
 Blackadar, Bruce. Operator Algebras. Encyclopedia of Mathematical Sciences. Vol. 122. Springer-Verlag. 2006.