Definition 34.10.1 (State).label Let $A$ be a unital $C^{*}$-algebra and $\phi \in A^{*}$, then $\phi$ is a state if $\phi$ is positive and $\dpn{1, \phi}{A}= 1$.

The set of states $S(A) \subset A^{*}$ of $A$ equipped with the weak* topology is the state space of $A$.