Choquet-Type Relations and a State Space Level Amendment of Arveson's Hyperrigidity Conjecture

Abstract

Davidson and Kennedy introduced the dilation order in connection with commutative C*-algebras and classical Choquet theory. Its noncommutative counterpart continues to detect the unique extension property of GNS representations. Motivated by the failure of Arveson's hyperrigidity conjecture and the amended theorem of Clou\^atre and Thompson, we develop a state-space approach to this rigidity phenomenon. First, we introduce the strong dilation relation on the state space of a C*-algebra and characterize the unique tight extension property through maximality of states. Second, we define the integral subdivision relation and prove that maximality of all pure states in the dilation order implies maximality of every state in the integral subdivision relation. This provides a state space-level amendment of Arveson's hyperrigidity conjecture and yields an alternative proof of the Clou\^atre-Thompson theorem. Finally, we show that the strong dilation and integral subdivision relations coincide in the commutative setting and they both agree with the abstract Choquet order associated with a function system.