State space of a C*-algebra

Let $A$ be a $C^*$ -algebra. Its state space, denoted $S(A)$ , is the set of all states on $A$ :

S(A)=\{\varphi\in A^*:\varphi\geq 0,\ \|\varphi\|=1\}.

It is a convex set in the Banach dual $A^*$ , equipped most often with the weak-star topology. If $A$ is unital, the norm condition is equivalent to $\varphi(1_A)=1$ , and $S(A)$ is weak-star compact. For nonunital $A$ , $S(A)$ need not be weak-star compact, although the positive functionals of norm at most one form a weak-star compact set.

Convex structure

A state is pure exactly when it is an extreme point of $S(A)$ . Thus convex decompositions express mixed states as combinations of other states. The Krein–Milman theorem implies that, for unital $A$ , the compact convex set $S(A)$ is the closed convex hull of its pure states.

Standard examples

For $A=C(X)$ with $X$ compact Hausdorff, $S(A)$ is the space of Radon probability measures on $X$ ; its pure states are the point masses. For $A=M_n(\mathbb C)$ , states correspond to positive matrices $\rho$ with $\operatorname{Tr}(\rho)=1$ through $\varphi(a)=\operatorname{Tr}(\rho a)$ . The pure states correspond to rank-one $\rho$ .

Topological caveat

References

Gerard J. Murphy, C-Algebras and Operator Theory*, Academic Press, 1990. DOI record. Relevant: §3.2 on states, convexity, and weak-star compactness.
Erik M. Alfsen and Frederic W. Shultz, State Spaces of Operator Algebras: Basic Theory, Orientations, and C-Products*, Birkhäuser, 2001. DOI record. Relevant: Part I on compact convex state spaces.