### 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 language | English |
---|---|

Pages (from-to) | 707-722 |

Number of pages | 15 |

Journal | Journal of the London Mathematical Society |

Volume | 78 |

Issue number | 3 |

DOIs | |

Publication status | Published - Aug 2008 |

### Keywords

- C-asterisk-algebras

### Cite this

*Journal of the London Mathematical Society*,

*78*(3), 707-722. https://doi.org/10.1112/jlms/jdn051

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

Research output: Contribution to journal › Article

*Journal of the London Mathematical Society*, vol. 78, no. 3, pp. 707-722. https://doi.org/10.1112/jlms/jdn051

}

TY - JOUR

T1 - Factorial states, upper multiplicity and norms of elementary operators

AU - Archbold, Robert J

AU - Somerset, Douglas W B

AU - Timoney, R. M.

PY - 2008/8

Y1 - 2008/8

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.

KW - C-asterisk-algebras

U2 - 10.1112/jlms/jdn051

DO - 10.1112/jlms/jdn051

M3 - Article

VL - 78

SP - 707

EP - 722

JO - Journal of the London Mathematical Society

JF - Journal of the London Mathematical Society

SN - 0024-6107

IS - 3

ER -