**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

for .

**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.

[1] Blackadar, Bruce. *Operator Algebras*. Encyclopedia of Mathematical Sciences. Vol. 122. Springer-Verlag. 2006.