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 ${\varphi }_1,{\varphi }_2,...$ from $\mathcal{B}(H)$ to itself, $$\underset{n\to \infty }{\lim }\Vert {\varphi }_n(g)-g\Vert =0,\forall g\in G⟹\underset{n\to \infty }{\lim }\Vert {\varphi }_n(a)-a\Vert =0,\forall a\in A.$$ 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 $G\cup G^{\ast }$. We present a variety of concrete applications and discuss prospects for further development.

A major revision, with new results in three new sections: substantial re-organization. 30 pages

Cited by in corpus (1)

• The noncommutative Choquet boundary III: Operator systems in matrix algebras