Resumen de Factorial states, upper multiplicity and norms of elementary operators

Robert J. Archbold, Douglas W. B. Somerset, Richard M. Timoney

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