= Exposed face of the dual ball defined by the unit
{title2=$S_A=\{x\in B_{A^*}:\operatorname{Re}x(1_A)=1\}$}
For a nonzero unital <C-star algebra>, a <positive functional on a C-star algebra> obeys $\|x\|=x(1_A)$, and conversely this equality for a bounded functional implies positivity. Therefore its <state on a C-star algebra> functionals are exactly the maximizers of the real linear evaluation $x\mapsto\operatorname{Re}x(1_A)$ on the closed dual unit ball. If a <convex combination> reaches that maximum, each constituent must reach it: each real part is at most one, and their weighted mean is one. The modulus bound then forces evaluation to equal one, so each constituent is again a state. Hence the state space is an <exposed face> and a <face of a convex set>.
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