Factorial states, upper multiplicity and norms of elementary operators

Abstract

Let {pi} 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 MU({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 use these localizations at ? 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.

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