Factorial states, upper multiplicity and norms of elementary operators

Abstract

Let π be an irreducible representation of a C*-algebra A. We show that the weak* approximation of factorial states associated to π by type I factorial states of lower degree is closely related to the value of the upper multiplicity MU(π) of π. 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 MU(π) > 1, then ‖ Tπ‖k ⩽ ‖ T‖n for certain k > n, and similarly that the n-positivity of T implies the k-positivity of Tπ (where Tπ is the induced operator on π(A)). We use these localizations at π 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.