  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.
