📝 SummarXiv

Abstract

We investigate the hyperrigidity of subsets of unital $C^*$-algebras annihilated by states (or, more generally, by completely positive maps). This is closely related to the concept of rigidity at $0$ introduced by G. Salomon, who studied hyperrigid subsets of Cuntz and Cuntz-Krieger algebras. The absence of the unit in a hyperrigid set allows for the existence of $R$-dilations with non-isometric $R$. The existence of such an $R$-dilation forces the state annihilating the hyperrigid set to be a character. Using a dilation-theoretic approach, we provide multiple equivalent criteria for hyperrigidity involving intertwining relations for representations, valid in both commutative and noncommutative settings. We develop structural models for such dilations via orthogonal decompositions into two or three components, determined by defect operators and generalized eigenspaces associated with underlying representations.