Factorial states, upper multiplicity and norms of elementary operators

Let p be an irreducible representation of a C*-algebra A. We show that the weak* approximation of factorial states associated to pi by type I factorial states of lower degree is closely related to the value of the upper multiplicity M-U(pi) of pi. As a consequence, we give a representation-theoretic characterization of those C*-algebras A for which the set of pure states P(A) is weak*-closed in the set of factorial states F(A). We also study the matricial norms and the positivity for elementary operators T on A. We show that if M-U(pi) > 1, then parallel to T-pi parallel to(k) <= parallel to T parallel to(n) for certain k > n, and similarly that the n-positivity of T implies the k-positivity of T-pi (where T-pi is the induced operator on pi(A)). We use these localizations at pi to give new proofs of various characterizations of the class of antiliminal-by-abelian C*-algebras in terms of factorial states and elementary operators. In the course of this, we show that antiliminal-by-abelian is equivalent to abelian-by-antiliminal.