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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.

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

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