Quasi hyperrigidity and weak peak points for non-commutative operator systems. (English) Zbl 1430.46042
W. Arveson [Isr. J. Math. 184, 349–385 (2011; Zbl 1266.46045)] initiated the study of the “noncommutative approximation theory” and dealt with the question of hyperrigidity of a set of generators of a \(C^*\)-algebra. The concept of non-commutative Choquet boundary and the relationship between hyperrigid operator systems and boundary representations was investigated thereof.
Motivated by this and Saskin’s result on Korovkin sets and Choquet boundary peak points, the authors introduce the weaker notions, namely, weak boundary representations, quasi hyperrigidity and weak peak points. They explore the relationship between weak Choquet boundary of an operator system \(S\), the set of all weak boundary representation, and quasi hyperrigidity. In the main result, the authors give a partial characterization of the weak Choquet boundary points that are also Choquet boundary points of an operator system.
Motivated by this and Saskin’s result on Korovkin sets and Choquet boundary peak points, the authors introduce the weaker notions, namely, weak boundary representations, quasi hyperrigidity and weak peak points. They explore the relationship between weak Choquet boundary of an operator system \(S\), the set of all weak boundary representation, and quasi hyperrigidity. In the main result, the authors give a partial characterization of the weak Choquet boundary points that are also Choquet boundary points of an operator system.
Reviewer: Preeti Luthra (Delhi)
MSC:
| 46L07 | Operator spaces and completely bounded maps |
| 46L52 | Noncommutative function spaces |
| 47L25 | Operator spaces (= matricially normed spaces) |
Keywords:
weak peak point; quasi hyperrigid set; weak boundary representation; boundary representationCitations:
Zbl 1266.46045References:
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