The noncommutative Choquet boundary II: Hyperrigidity

Details The noncommutative Choquet boundary II: Hyperrigidity The noncommutative Choquet boundary II: Hyperrigidity

2008-10-15

arxiv.org/abs/0810.2751v4

Mathematics

Operator Algebras

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

Scientific paper

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 $\phi_1, \phi_2,...$ from $\mathcal B(H)$ to itself, $$ \lim_{n\to\infty}\|\phi_n(g)-g\|=0, \forall g\in G \implies \lim_{n\to\infty}\|\phi_n(a)-a\|=0, \forall a\in A. $$ 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.

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