States and Representations of C*-algebras

Definition 1.  Let A be a C*-algebra and x\in Ax 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.

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