The noncommutative Choquet boundary II: Hyperrigidity

Abstract

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 $ϕ_1, ϕ_2,...$ from $\mathcal B(H)$ to itself, $$ \lim_{n\to\infty}\|ϕ_n(g)-g\|=0, \forall g\in G \implies \lim_{n\to\infty}\|ϕ_n(a)-a\|=0, \forall a\in A. $$ 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 $G\cup G^*$. We present a variety of concrete applications and discuss prospects for further development.