# States and Representations of C*-algebras

Definition 1.  Let $A$ be a C*-algebra and $x\in A$$x$ is positive if it is self-adjoint and $\sigma_A(x)\subseteq [0,\infty)$.  We denote the set of positive elements of $A$ by $A_+$ and write $x\geq 0$ for all $x\in A_+$.  We also write $x\leq y$ if $y-x\geq 0$.

It turns out that if $B$ is a C*-subalgebra of $A$, then $\sigma_B(x)=\sigma_A(x)$.  Hence $B_+=B\cap A_+$.

Definition 2.  A functional $f:A\to\mathbb{C}$ is positive if $x\geq 0\Rightarrow f(x)\geq 0$ (hence $f(x)\in\mathbb{R}$).  A state is a positive linear functional such that $\|f\|=\sup_{x\in A}|f(x)|=1$.  The set of all states is called the state space and is denoted $S(A)$.

If $f$ is a positive linear functional, it defines a pre inner product on $A$:

$\langle x,y\rangle_f=f(y^*x)$.

If $H$ is a Hilbert space, we can endow a *-algebra structure on $L(H)$, the space of linear operators on $H$ where multiplication is composition, and if $X\in L(H)$, then $X^*$ is defined as the adjoint of $X$.

Definition 3.  Let $A$ be a C*-algebra and $H$ be a Hilbert space.  A representation is a *-homomorphism $\rho:A\to L(H)$.  A subrepresentation is a representation $\rho':A\to L(H')$ where $H'\leq H$ is a closed subspace of $H$ which is invariant under action from $A$.  A representation is irreducible if it has no nontrivial subrepresentations.  A representation $\rho$ is faithful if $\ker\rho=0$.

We now present an important connection discovered by Gelfand, Naimark, and Segal.

Let $A$ be a C*-algebra and $f$ be a positive linear functional on $A$.  Define

$N_f=\{x\in A:f(x^*x)=0\}$.

$N_f$ is a closed left ideal in $A$, and $(A/N_f,\langle\cdot\rangle_f)$ is an inner product space.  Let $H_f=L^2(A,f)$ be its Hilbert completion.  Now define $\rho_f:A\to L(H_f)$ by action of $x$: $\rho_f(x)(a+N_f)=xa+N_f$$\rho_f$ is clearly a representation, and is called the GNS representation of $A$ associated with $f$.

If $A$ is unital, let $1_f$ denote the image of $1$ in the completion/quotient composition $A\to A/N_f\to H_f$.  Then $f$ induces a positive linear functional on $L(H_f)$ defined by

$\phi_f(X)=\langle X(1_f),1_f\rangle_f$

for $X\in L(H_f)$.

Theorem 4.  Let $f,g$ be positive linear functionals on a C*-algebra $A$ such that $g\leq f$ (meaning $g(x)\leq f(x)$ for all $x\in A_+$).  Then there is a unique operator $X\in\rho_f(A)\subseteq L(H_f)$ such that $0\leq X\leq 1$ and $g(x)=\phi_f(X\rho_f(x))=\langle (X\rho_f(x)(1_f),1_f\rangle_f$ for all $x\in A$.

This is a generalization of the Radon-Nikodym theorem (which is special case for $A=L^\infty(X,\mu)$ where $(X,\mu)$ is a finite measure space and $\phi(f)=\int f\,d\mu$).

Definition 5.  We define an extreme point in topological vector space $V$ as a point that does not belong to any open line segment in $V$.  A pure state is an extreme point in $S(A)$.  The set of pure states of $A$ is denoted $P(A)$.

It follows that $S(A)=hull(P(A))$.

Proposition 6.  Let $f$ be a state on a C*-algebra $A$.  Then $\rho_f$ is irreducible if and only if $f$ is pure.

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