Rigidity of operator systems: tight extensions and noncommutative measurable structures
Raphaël Clouâtre, Ian Thompson
TL;DR
The paper reframes Arveson's hyperrigidity conjecture through noncommutative measurable structures and tight extensions, proving that for separable operator systems $S$, every irreducible $*$-representation of $A$ is a boundary representation for $S$ if and only if every unital $*$-representation has a unique tight extension with respect to $S$. It extends Korovkin–Šaškin rigidity to the noncommutative setting, showing that in nuclear (and, in homogeneous cases, strongly aligned with the identity) scenarios, boundary representations, and Korovkin rigidity properties for representations are equivalent. A detailed analysis of strong Korovkin rigidity reveals that, under lifting properties and homogenity, norm convergence of CP-approximants on $S$ extends to all of $A$, with a corresponding invariance under representation changes. The Bilich–Dor-On counterexample is reconciled within this amended framework, demonstrating that the new notions of unique tight extension and Korovkin rigidity hold, thereby salvaging a robust form of rigidity for operator systems even when hyperrigidity fails.
Abstract
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.