Given an operator system S there are two sequences of new operator
systems that one can construct from S. One sequence is universal for
K-positive maps with range S and the other for K-positive maps with domain
S. We prove that a certain convergence of these new systems to S as K tends
to infinity imply that the operator system is either exact or has a local
lifting property, depending on the case. For finite dimensional operator
systems we can prove the converse of these results using matrix ranges,
which are a dual for operator systems. This talk is based on joint work with
K. Davidson, B. Passer and M. Rahaman.
