### Abstract

For vector functionals on a C*-algebra of operators, we prove an analogue of Glimm's vector state space theorem. We deduce that a C*-algebra is prime and antiliminal if and only if the pure functionals are w*-dense in the unit ball of the dual. We also give a necessary and sufficient condition for a convex combination of inequivalent pure functionals to be a w*-limit of pure functionals.

Original language | English |
---|---|

Pages (from-to) | 39-51 |

Number of pages | 12 |

Journal | Journal of Operator Theory |

Volume | 45 |

Publication status | Published - 2001 |

### Keywords

- C*-algebra
- vector functional
- pure functional
- antiliminal
- prime
- PURE STATES

### Cite this

*Journal of Operator Theory*,

*45*, 39-51.

**Limits of vector functionals.** / Archbold, Robert J; Shah, M. H.

Research output: Contribution to journal › Article

*Journal of Operator Theory*, vol. 45, pp. 39-51.

}

TY - JOUR

T1 - Limits of vector functionals

AU - Archbold, Robert J

AU - Shah, M. H.

PY - 2001

Y1 - 2001

N2 - For vector functionals on a C*-algebra of operators, we prove an analogue of Glimm's vector state space theorem. We deduce that a C*-algebra is prime and antiliminal if and only if the pure functionals are w*-dense in the unit ball of the dual. We also give a necessary and sufficient condition for a convex combination of inequivalent pure functionals to be a w*-limit of pure functionals.

AB - For vector functionals on a C*-algebra of operators, we prove an analogue of Glimm's vector state space theorem. We deduce that a C*-algebra is prime and antiliminal if and only if the pure functionals are w*-dense in the unit ball of the dual. We also give a necessary and sufficient condition for a convex combination of inequivalent pure functionals to be a w*-limit of pure functionals.

KW - C-algebra

KW - vector functional

KW - pure functional

KW - antiliminal

KW - prime

KW - PURE STATES

M3 - Article

VL - 45

SP - 39

EP - 51

JO - Journal of Operator Theory

JF - Journal of Operator Theory

SN - 0379-4024

ER -