According to the article on nLab the Alfsen Shultz theorem states that the space of states of a given $C^*$-algebra depends on somehow weaker structure namely on the so called *Jordan algebra* structure. This article gives reference for this theorem: however I've checked the cited paper and I don;t see how the main theorem (namely theorem 9.5 in this paper) implies the above statement. For convienience I quote this theorem below: Let $A$ be a JB algebra (see the Introduction of the paper). Then there is unique Jordan ideal $J$ such that $A/J$ has faithful, isometric Jordan representation as an JC algebra and every factor representation of $A$ not anihilating $J$ is onto the exceptional algebra $M_3^8$ .

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strongly spectral, a notion defined here, although perhaps the place to look is the two books by Alfsen and Shultz calledState Spaces of Operator AlgebrasandGeometry of State Spaces of Operator Algebras. $\endgroup$ – Robert Furber Aug 16 '20 at 0:37i.e.there is a $y$ such that $x = y \circ y$. (This agrees with the fact that positive elements of C$^*$-algebras are the squares of self-adjoint elements.) The positive cone and unit allow you to define when a linear functional is a state, and get the state space. $\endgroup$ – Robert Furber Aug 16 '20 at 21:10