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

A subset $\mathcal{G}$ generating a $C^{\ast }$-algebra $A$ is said to be hyperrigid if for every faithful nondegenerate $\ast$-representation $A\subseteq B(H)$andasequence$\phi_n:B(H) \to B(H)$ of unital completely positive maps, we have that $$\underset{n\to \mathrm{\infty }}{\lim }{\varphi }_n(g)=g\text{ for all }g\in \mathcal{G}⟹\underset{n\to \mathrm{\infty }}{\lim }{\varphi }_n(a)=a\text{ for all }a\in A$$ 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 $C^{\ast }$-algebra $A$ is said to be rigid at $0$ if for every sequence of contractive positive maps ${\phi }_n:A\to \mathbb{C}$ satisfying ${\lim }_{n\to \mathrm{\infty }}{\phi }_n(g)=0$ for every $g\in \mathcal{G}$, we have that ${\lim }_{n\to \mathrm{\infty }}{\phi }_n(a)=0$ 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.