State on a C*-algebra - Imported Knowlpedia Content

Definition

Let $A$ be a $C^*$ -algebra. A state on $A$ is a positive linear functional $\varphi:A\to\mathbb C$ satisfying $\|\varphi\|=1$ . Thus a state is bounded and norm-continuous by definition, and positivity means $\varphi(a^*a)\geq0$ for every $a\in A$ . If $A$ is unital, the normalization is equivalent to $\varphi(1)=1$ . For a nonunital algebra the norm condition remains the definition; no multiplier-unit value is silently assumed. The set of all states is denoted $S(A)$ . Normality is additional data available when $A$ is a von Neumann algebra, not part of a general $C^*$ -state.

Convex structure

The state space $S(A)$ is convex: if $\varphi,\psi\in S(A)$ and $0\leq t\leq1$ , then $t\varphi+(1-t)\psi$ is again a state. A state is pure when it is an extreme point of this convex set. Purity is therefore not part of the definition of a state. For unital $A$ , the state space is weak-star compact in $A^*$ ; for nonunital $A$ , states may have weak-star limits of norm less than one, so $S(A)$ need not be weak-star closed.

Representation-theoretic meaning

The GNS construction writes every state as a vector functional

\varphi(a)=\langle\pi_\varphi(a)\xi_\varphi,\xi_\varphi\rangle

for a cyclic representation with $\|\xi_\varphi\|=1$ . The state is pure exactly when its GNS representation is irreducible Murphy, §3.3. Faithfulness of a state and irreducibility of its representation are different properties: neither implies the other in general.

Examples and distinctions

For $A=B(H)$ , each unit vector $\xi\in H$ defines the vector state $\varphi_\xi(T)=\langle T\xi,\xi\rangle$ . A density operator $\rho$ defines the normal state $\varphi_\rho(T)=\operatorname{Tr}(\rho T)$ . On a commutative unital algebra $C(X)$ , evaluation $f\mapsto f(x)$ is a pure state, whereas integration against a probability measure is a general state. A tracial state additionally satisfies $\varphi(ab)=\varphi(ba)$ ; most states are not tracial.

References

Gerard J. Murphy, $C^*$ -Algebras and Operator Theory, Academic Press, 1990. DOI record. Relevant: §§3.2–3.3 on states, pure states, and GNS representations.
Gert K. Pedersen, $C^*$ -Algebras and Their Automorphism Groups, 2nd ed., Academic Press, 2018. DOI record. Relevant: the chapter on positive functionals and state spaces.