 State on a C-star algebra

A state on a C-star algebra is a positive functional on a C-star algebra of norm one. For a nonzero unital C-star algebra $A$, its state space $S(A)$ is a compact for the weak-star topology convex set: positivity and $f(1)=1$ define a closed for the weak-star topology subset of the dual unit ball, and the Banach-Alaoglu theorem gives compactness.

For a nonunital C-star algebra, the state space need not be compact for the weak-star topology. In $C_0(\mathbb{R})$, evaluations ${\delta }_n(f)=f(n)$ are states, but $f(n)\to 0$ for every $f\in C_0(\mathbb{R})$. Thus ${\delta }_n$ converges in the weak-star topology to the zero functional, which has norm zero and is not a state.