Korovkin-type properties for completely positive maps

We prove a noncommutative variant of Saskin's classical theorem -- on the connection between Choquet boundaries for function spaces and Korovkin sets -- for operator systems generating separable Type I C*-algebras. The main result implies an affirmative answer to a weaker version of Arveson's hyperrigidity conjecture for such C*-algebras. It also yields information about the more general version of Arveson's conjecture.