By a result of Kadison, every operator subsystem of a commutative $C^*$-algebra is isomorphic to the space of continuous affine functions on its state space.

In other words, if $X$ is a compact Hausdorff space and $E\subseteq C(X)$ is an operator system, then $Aff(S(E))$ is isomorphic to $E$ (as operator systems- complete order isomorphism), where $S(E)$ denotes the state space.

This seems to be a generalisation of Gelfand Duality for commutative $C^*$-algebras.

I would be happy to get a recommended reference for a proof.

Moreover, I could not see why in the case that $E$ is a unital, commutative $C^*$-algebra, i.e. $E=C(X)$, then we actually get $A(S(C(X)))=C(X)$ as expected? Since $S(C(X))$ can be identified with the space of Radon probability measures on $X$, $P(X)$, my question reduces to the question: Why the continuous affine functions on $P(X)$ are isomorphic to $C(X)$?


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    $\begingroup$ Hint: You can use that the extreme points of $\mathbb P X$ are the Dirac deltas $(\delta_x)_{x \in X}$ (for this you will need to connect the extreme points of the state space with irreducible representations). $\endgroup$ – Adrián González-Pérez Nov 3 '18 at 22:43
  • $\begingroup$ Thanks! Of course, this answers my second question. $\endgroup$ – Shirly Geffen Nov 3 '18 at 22:56

You can check Chapter II of Compact convex sets and boundary integrals by Alfsen


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