Definition 32.3.2 (State).label Let $G$ be a locally compact group and $\phi \in L^{1}(G; \complex)^{*}$ be a positive linear functional, then $\phi$ is a state if $\norm{\phi}_{L^1(G; \complex)^*}= 1$. The set $S(G)$ of all states on $L^{1}(G; \complex)$ is the state space of $G$.