We prove that an operator system S is nuclear in the category of operator systems if and only if there exist nets of unital completely positive maps φλ:S→Mnλ and ψλ:Mnλ→S such that ψλ∘φλ converges to idS in the point-norm topology.
By compactness, there exist nets ({p_{W}}) converging to some (bar{p}in P), (r(p_{W} rightarrow r(bar{p})=bar{x}) in X, (x_{W}rightarrowbar{x}) in X, (y_{W}rightarrowbar{y}) in (operatorname{cl}(Gamma(X))), (s y_{W} rightarrow s(bar{y})) in P. Consequently, (bar{p}=s(bar{y})in sGamma(bar {x})=sGamma r(bar{p})).
By the lower semicontinuity of and Lemma 2.5 iii), for any and any net, there exists a net such that and.
By the lower semicontinuity of and Lemma 2.7 iii), we note that for any and any net, there exists a net such that and.
Since is a lower semicontinuous, it follows by Lemma 2.7 iii) that for any and any net, there exists a net such that and.
(iii) is lower semicontinuous at if and only if for any and any net, there exists a net such that and.
Then, the following properties hold: (i if is closed and is compact, then is upper semicontinuous, where and denotes the closure of the set, (ii)if is upper semicontinuous and for any is closed, then is closed, (iii) is lower semicontinuous at if and only if for any and any net, there exists a net such that and.
Then is l.s.c. at if and only if for any and any net in converges to, there exists a net such that for all and.
Then, is l.s.c. at if and only if for any and for any net in converging to, there exists a net in such that for each and converges to.
