# Journal of Operator Theory

Volume 81, Issue 1, Winter 2019 pp. 61-79.

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

**Authors**:
Guy Salomon

**Author institution:** Pure Mathematics Department, University of Waterloo,
Waterloo, Ontario N2L 3G1, Canada

**Summary: ** A subset $\mathcal{G}$ generating a $C^*$-algebra $A$ is said
to be \textit{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
\[
\lim_{n\to\infty}\phi_n(g)= g\quad\text{for all } g\in \mathcal{G} \implies
\lim_{n\to\infty}\phi_n(a)= a\quad\text{for all } a\in A.
\]
We show that in the Cuntz-Krieger algebra of a row-finite directed graph
with no isolated vertices, the set of all edge partial-isometries
is hyperrigid. We also examine, both in general and in the context of
graphs, a related property named \textit{rigidity at} 0 that sheds light on
the phenomenon of hyperrigidity.

**DOI: **http://dx.doi.org/10.7900/jot.2017nov02.2197

**Keywords: ** Cuntz-Krieger algebra, directed graph, hyperrigidity, $C^*$-envelope

Contents
Full-Text PDF