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

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.