Hridoyananda Saikia | Searcharxiv

arXiv2026

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

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 Clouâtre 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 Clouâtre-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.

math.OA↗

arXiv2023

A boundary projection for the dilation order

Motivated by Arveson's conjecture, we introduce a notion of hyperrigidity for a partial order on the state space of a $C^*$-algebra $B$. We show how this property is equivalent to the existence of a boundary: a subset of the pure states which completely encodes maximality in the given order. In the classical case where $B$ is commutative, such boundaries are known to exist when the partial order is induced by some well-behaved cone. However, the relevant order for the purposes of Arveson's conjecture is the dilation order, which is not known to fit into this framework. Our main result addresses this difficulty by showing that the dilation maximal states are stable under absolute continuity. Consequently, we obtain the existence of a boundary projection in the bidual $B^{**}$, on which all dilation maximal states must be concentrated. The topological regularity of this boundary projection is shown to lie at the heart of Arveson's conjecture. Our techniques do not require $B$ to be commutative.

math.OA↗