Weak Korovkin approximation by completely positive linear maps on β(H)

AbstractA subset l of β(H) is called a weak Korovkin set if, for every net (φα) of complryrly positive linear maps on β(H) with φα(I)≤I, the relation φα(S) → S weakly, S ϵl, implies φα(T) → T weakly, T ϵ β(H). Let l be an irreducilbe subset of β(H) which contains the identity operator.Theorem. (i) If there are T ϵ spanl + l∗ and a compact operator K such that ∥T - K∥ < ∥T∥, then l is a weak Korovkin set. (ii) If C∗(l) contains a nonzero compact operator, then f = S:S ϵ l ⊂ S∗S + SS∗:S ϵ l is a weak Korovkin set. In particular, if l = I,S1,…Sm, then f1 = I, S1,…,Sm, Σj=1m (Sj∗+SjSj∗) is a weak Korovkin set