The noncommutative Choquet boundary II: Hyperrigidity

TL;DR

This paper investigates hyperrigidity of generator sets in C*-algebras using noncommutative Choquet boundary analysis, providing criteria and applications for identifying hyperrigid sets.

Contribution

It introduces a method to determine hyperrigidity via the noncommutative Choquet boundary and explores various applications and future directions.

Findings

Hyperrigidity can be characterized by the noncommutative Choquet boundary.

A practical criterion for checking hyperrigidity is provided.

The paper discusses multiple applications and potential developments.

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 $\phi_1, \phi_2,...$ 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.