Non-commutative peaking phenomena and a local version of the hyperrigidity conjecture

TL;DR

This paper explores peaking phenomena in $ ext{C}^*$-algebras within operator systems, introducing a localization technique to verify a variation of Arveson's hyperrigidity conjecture.

Contribution

It develops a new approach to localize $ ext{C}^*$-algebras at states using characteristic sequences, advancing the understanding of hyperrigidity in operator systems.

Findings

Established a local version of the hyperrigidity conjecture for operator systems.

Introduced characteristic sequences to approximate point evaluations in state spaces.

Provided new tools for analyzing peaking phenomena in $ ext{C}^*$-algebras.

Abstract

We investigate various notions of peaking behaviour for states on a $\mathrm{C}^*$-algebra, where the peaking occurs within an operator system. We pay particularly close attention to the existence of sequences of elements forming an approximation of the characteristic function of a point in the state space. We exploit such characteristic sequences to localize the $\mathrm{C}^*$-algebra at a given state, and use this localization procedure to verify a variation of Arveson's hyperrigidity conjecture for arbitrary operator systems.