Rigidity of operator systems: tight extensions and noncommutative measurable structures

TL;DR

This paper revises Arveson's conjecture on hyperrigidity of operator systems within $C^*$-algebras, linking boundary representations to unique tight extensions and noncommutative measurable structures.

Contribution

It establishes a new equivalence between boundary representations and tight extensions, extending classical approximation principles to noncommutative settings.

Findings

All irreducible representations are boundary representations iff all admit a unique tight extension.

Proves equivalence between tight extension uniqueness and rigidity of CP approximations.

Extends classical Korovkin--Saskin principle to nuclear $C^*$-algebras.

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.