2024/06/24 by Raphaël Clouâtre, Ian M. Thompson, Clouâtre, Raphaël +1 · 1 citation
Mathematics · #Advanced Operator Algebra Research #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Mathematical Analysis and Transform Methods #Operator Algebras (math.OA)
paper · pdf · doi:10.48550/arxiv.2406.16806
openalex publication_date 2024/06/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30
Let A be a unital C^*-algebra generated by some separable operator system S. More than a decade ago, Arveson conjectured that S is hyperrigid in A if all irreducible representations of A are boundary representations for S. Recently, a counterexample to the conjecture was found by Bilich and Dor-On. To circumvent the difficulties hidden in this counterexample, we exploit some of Pedersen's seminal ideas on noncommutative measurable structures and establish an amended version of Arveson's conjecture. More precisely, we show that all irreducible representations of A are boundary representations for S precisely when all representations of A admit a unique "tight" completely positive extension from S. In addition, we prove an equivalence between uniqueness of such tight extensions and rigidity of completely positive approximations for representations of nuclear C^*-algebras, thereby extending the classical principle of Korovkin--Saskin for commutative algebras of continuous functions.