### 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 |
---|---|

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

### Cite this

**Extensions of pure states and projections of norm one.** / Archbold, Robert J.

Research output: Contribution to journal › Article

*Journal of Functional Analysis*, vol. 165, pp. 24-43.

}

TY - JOUR

T1 - Extensions of pure states and projections of norm one

AU - Archbold, Robert J

PY - 1999

Y1 - 1999

N2 - 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.

AB - 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.

KW - C-algebras

KW - pure state

KW - unique extension

KW - projection of norm one

KW - AF algebra

KW - ultraliminary

KW - C-ASTERISK-ALGEBRAS

KW - OPERATOR-ALGEBRAS

KW - STAR-ALGEBRAS

KW - FACTORIAL STATES

KW - CSTAR-ALGEBRAS

KW - REPRESENTATIONS

KW - MULTIPLICITY

KW - PRODUCTS

KW - TRACE

KW - MAPS

M3 - Article

VL - 165

SP - 24

EP - 43

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

ER -