Arveson’s hyperrigidity conjecture concerns the unique extension property of -representations of a C-algebra with respect to a generating operator system. The maximal states in the dilation order fully encapsulate the cyclic representations of a C*-algebra with the unique extension property. A reformulation of the conjecture by Davidson and Kennedy raises the question whether the maximal measures in the dilation order are concentrated on a particular set. In this thesis, we address this question for general C*-algebras. We show the existence of a projection such that the dilation maximal states are precisely those states which are concentrated on the projection. We also reformulate the conjecture in terms of the non-commutative topological properties of this projection.

Choquet order is a partial order defined on the set of regular Borel probability measures on a compact convex set. With the help of two equivalent characterizations of Choquet order, we define strong dilation relation and sub-division relation on the state space of a C*-algebra. The equivalence of the two relations is not known in general. We show that the strong dilation relation is stronger than the sub-division relation. Moreover, we show the equivalence of the strong dilation relation with a non-commutative sub-division relation. We also demonstrate that these relations can serve as valuable tools for investigating certain rigidity properties of a generating operator system of a C*-algebra.