Hyperrigid subsets of Cuntz-Krieger algebras and the property of rigidity at zero

TL;DR

This paper studies hyperrigidity and rigidity at zero in Cuntz-Krieger algebras, establishing conditions under which generating sets exhibit these properties and their implications for the algebra's structure.

Contribution

It introduces and analyzes the concepts of hyperrigidity and rigidity at zero for subsets of Cuntz-Krieger algebras, linking these properties to the unique extension property and graph structure.

Findings

The set of partial isometries in Cuntz-Krieger algebras is hyperrigid.

Rigidity at zero for generating subsets occurs if and only if they include all vertex projections.

Hyperrigidity combined with rigidity at zero implies a stronger form of hyperrigidity.

Abstract

A subset $\mathcal{G}$ generating a $C^*$-algebra $A$ is said to be hyperrigid if for every faithful nondegenerate $*$-representation $A\subseteq B(H)$ and a sequence $\phi_n:B(H) \to B(H)$ of unital completely positive maps, we have that \[ \lim_{n\to\infty}\phi_n(g)= g~~\text{for all } g\in \mathcal{G} ~~ \implies ~~ \lim_{n\to\infty}\phi_n(a)= a~~\text{for all } a\in A \] where all convergence are in norm. In this paper, we show that for the Cuntz-Krieger algebra $\mathcal{O}(G)$ associated to a row-finite directed graph $G$ with no isolated vertices, the set of partial isometries $\mathcal{E}=\{S_e:e\in E\}$ is hyperrigid. In addition, we define and examine a closely related notion: the property of rigidity at $0$. A generating subset $\mathcal{G}$ of a $C^*$-algebra $A$ is said to be rigid at $0$ if for every sequence of contractive positive maps $\varphi_n:A\to \mathbb C$ satisfying $\lim_{n\to \infty}\varphi_n(g)=0$ for every $g\in \mathcal{G}$, we have that $\lim_{n\to \infty}\varphi_n(a)=0$ for every $a\in A$. We show that, when combined, hyperrigidity and rigidity at $0$ are equivalent to a somewhat stronger notion of hyperrigidity, and we connect this to the unique extension property. This, however, is not the case for the generating set $\mathcal{E}$. More precisely, we show that for any graph $G$, subsets of the Cuntz-Krieger family generating $\mathcal{O}(G)$ are rigid at $0$ if and only if they contain every vertex projection.