Factorial states, upper multiplicity and norms of elementary operators

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

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)707-722
Number of pages15
JournalJournal of the London Mathematical Society
Volume78
Issue number3
DOIs
Publication statusPublished - Aug 2008

Keywords

  • C-asterisk-algebras

Cite this

Factorial states, upper multiplicity and norms of elementary operators. / Archbold, Robert J; Somerset, Douglas W B; Timoney, R. M.

In: Journal of the London Mathematical Society, Vol. 78, No. 3, 08.2008, p. 707-722.

Research output: Contribution to journalArticle

Archbold, Robert J ; Somerset, Douglas W B ; Timoney, R. M. / Factorial states, upper multiplicity and norms of elementary operators. In: Journal of the London Mathematical Society. 2008 ; Vol. 78, No. 3. pp. 707-722.
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N2 - 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.

AB - 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.

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