### Abstract

We show that the extension property for pure states of a C*-subalgebra B of a C*-algebra. A leads to the existence of a projection of norm one R: A --> B in the case where B is liminal with Hausdorff primitive ideal space. Furthermore, R is given by a "Dixmier process" in which the averaging is affected by a group of unitary elements in the centre of the multiplier algebra M(B). These results generalize earlier work of J. Anderson and the author for the case when B is a masa of a. Various applications are given in the contest of inductive limit algebras such as AF algebras and, more generally, Kumjian's ultraliminary C*-algebras. (C) 1999 Academic Press.

Original language | English |
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Pages (from-to) | 24-43 |

Number of pages | 20 |

Journal | Journal of Functional Analysis |

Volume | 165 |

Publication status | Published - 1999 |

### Keywords

- C*-algebras
- pure state
- unique extension
- projection of norm one
- AF algebra
- ultraliminary
- C-ASTERISK-ALGEBRAS
- OPERATOR-ALGEBRAS
- STAR-ALGEBRAS
- FACTORIAL STATES
- CSTAR-ALGEBRAS
- REPRESENTATIONS
- MULTIPLICITY
- PRODUCTS
- TRACE
- MAPS