The noncommutative Choquet boundary II: Hyperrigidity

A (finite or countably infinite) set G of generators of an abstract C*-algebra A is called hyperrigid if for every faithful representation of A on a Hilbert space $A\subseteq \mathcal{B}(H)$ and every sequence of unital completely positive linear maps ${\varphi }_1,{\varphi }_2,\mathrm{.}\mathrm{.}\mathrm{.}$ from $\mathcal{B}(H)$ to itself, $$\underset{n\to \mathrm{\infty }}{\lim }\mathrm{\parallel }{\varphi }_n(g)-g\mathrm{\parallel }=0,\mathrm{\forall }g\in G⟹\underset{n\to \mathrm{\infty }}{\lim }\mathrm{\parallel }{\varphi }_n(a)-a\mathrm{\parallel }=0,\mathrm{\forall }a\in A\mathrm{.}$$ We show that one can determine whether a given set G of generators is hyperrigid by examining the noncommutative Choquet boundary of the operator space spanned by $G\cup G^{\ast }$. We present a variety of concrete applications and discuss prospects for further development.