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

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.