Positive functional on a C-star algebra

A bounded linear functional $\tau$ on a C-star algebra is positive when $\tau (x)\ge 0$ for every positive element of a C-star algebra. Positivity impliesand $\Vert \tau \Vert =\tau (1)$; conversely, $\Vert \tau \Vert =\tau (1)$ implies positivity.

A state is a positive functional of norm one. The state space $S(A)$ is a weak-star compact convex subset of $A^{\ast }$.

A pure state is an extreme point of the state space. The Krein-Milman theorem guarantees pure states, and every positive element attains its norm at some pure state.