Title: 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.