2023/10/26 by Raphaël Clouâtre, Clouâtre, Raphaël, Hridoyananda Saikia +1 · 1 citation
Mathematics · #Advanced Operator Algebra Research #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA)
paper · pdf · doi:10.48550/arxiv.2310.17601
openalex publication_date 2023/10/26 · openalex created_date 2023/10/28 · openalex updated_date 2026/07/28
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.