Hyperrigidity II: $R$-dilations, ideals and decompositions

TL;DR

This paper explores the concept of hyperrigidity in unital C*-algebras, establishing criteria and models for R-dilations and their relation to states and ideals, advancing the understanding of rigidity phenomena in operator algebras.

Contribution

It introduces new dilation-based criteria for hyperrigidity, characterizes when R-dilations exist with non-isometric R, and develops structural models for such dilations.

Findings

Criteria for hyperrigidity involving intertwining relations

Existence of R-dilations linked to character states

Structural models using orthogonal decompositions

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.