# Kadison's Theorem for operator subsystems of commutative $C^*$-algebras

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 $$C(X)$$?

Thanks!

• 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). – Adrián González-Pérez Nov 3 '18 at 22:43
• Thanks! Of course, this answers my second question. – Shirly Geffen Nov 3 '18 at 22:56