### Abstract

Let A be a stable, sigma-unital, continuous C-0(X)-algebra with surjective base map phi : Prim(A) -> X, where Prim(A) is the primitive ideal space of the C*-algebra A. Suppose that phi(-1) (x) is contained in a limit set in Prim(A) for each x is an element of X (so that A is quasi-standard). Let C-R(X) be the ring of continuous real-valued functions on X. It is shown that there is a homeomorphism between the space of minimal prime ideals of C-R(X) and the space MinPrimal(M(A)) of minimal closed primal ideals of the multiplier algebra M(A). If A is separable then MinPrimal(M(A)) is compact and extremally disconnected but if X = beta N \ N then MinPrimal(M(A)) is nowhere locally compact.

Original language | English |
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Title of host publication | Operator Semigroups Meet Complex Analysis, Harmonic Analysis and Mathematical Physics |

Editors | Wolfgang Arendt, Ralph Chill, Yuri Tomilov |

Publisher | Springer |

Pages | 17-29 |

Number of pages | 13 |

ISBN (Electronic) | 978-3-319-18494-4 |

ISBN (Print) | 978-3-319-18493-7 |

DOIs | |

Publication status | Published - 2015 |

Event | Conference on Operator Semigroups Meet Complex Analysis, Harmonic Analysis and Mathematical Physics - Herrenhut, Germany Duration: 1 Jun 2013 → … |

### Publication series

Name | Operator Theory: Advances and Applications |
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Publisher | Birkhäuser Basel |

Volume | 250 |

ISSN (Print) | 0255-0156 |

ISSN (Electronic) | 0255-0156 |

### Conference

Conference | Conference on Operator Semigroups Meet Complex Analysis, Harmonic Analysis and Mathematical Physics |
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Country | Germany |

City | Herrenhut |

Period | 1/06/13 → … |

### Fingerprint

### Keywords

- C∗-algebra
- C0(X)-algebra
- multiplier algebra
- minimal prime ideal
- minimal primal ideal
- primitive ideal space
- quasi standard

### Cite this

_{0}(

*X*)-algebra. In W. Arendt, R. Chill, & Y. Tomilov (Eds.),

*Operator Semigroups Meet Complex Analysis, Harmonic Analysis and Mathematical Physics*(pp. 17-29). (Operator Theory: Advances and Applications; Vol. 250). Springer . https://doi.org/10.1007/978-3-319-18494-4_2

**Minimal Primal Ideals in the Multiplier Algebra of a C _{0}(X)-algebra.** / Archbold, R. J.; Somerset, D. W. B.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

_{0}(

*X*)-algebra. in W Arendt, R Chill & Y Tomilov (eds),

*Operator Semigroups Meet Complex Analysis, Harmonic Analysis and Mathematical Physics.*Operator Theory: Advances and Applications, vol. 250, Springer , pp. 17-29, Conference on Operator Semigroups Meet Complex Analysis, Harmonic Analysis and Mathematical Physics, Herrenhut, Germany, 1/06/13. https://doi.org/10.1007/978-3-319-18494-4_2

_{0}(

*X*)-algebra. In Arendt W, Chill R, Tomilov Y, editors, Operator Semigroups Meet Complex Analysis, Harmonic Analysis and Mathematical Physics. Springer . 2015. p. 17-29. (Operator Theory: Advances and Applications). https://doi.org/10.1007/978-3-319-18494-4_2

}

TY - GEN

T1 - Minimal Primal Ideals in the Multiplier Algebra of a C0(X)-algebra

AU - Archbold, R. J.

AU - Somerset, D. W. B.

PY - 2015

Y1 - 2015

N2 - Let A be a stable, sigma-unital, continuous C-0(X)-algebra with surjective base map phi : Prim(A) -> X, where Prim(A) is the primitive ideal space of the C*-algebra A. Suppose that phi(-1) (x) is contained in a limit set in Prim(A) for each x is an element of X (so that A is quasi-standard). Let C-R(X) be the ring of continuous real-valued functions on X. It is shown that there is a homeomorphism between the space of minimal prime ideals of C-R(X) and the space MinPrimal(M(A)) of minimal closed primal ideals of the multiplier algebra M(A). If A is separable then MinPrimal(M(A)) is compact and extremally disconnected but if X = beta N \ N then MinPrimal(M(A)) is nowhere locally compact.

AB - Let A be a stable, sigma-unital, continuous C-0(X)-algebra with surjective base map phi : Prim(A) -> X, where Prim(A) is the primitive ideal space of the C*-algebra A. Suppose that phi(-1) (x) is contained in a limit set in Prim(A) for each x is an element of X (so that A is quasi-standard). Let C-R(X) be the ring of continuous real-valued functions on X. It is shown that there is a homeomorphism between the space of minimal prime ideals of C-R(X) and the space MinPrimal(M(A)) of minimal closed primal ideals of the multiplier algebra M(A). If A is separable then MinPrimal(M(A)) is compact and extremally disconnected but if X = beta N \ N then MinPrimal(M(A)) is nowhere locally compact.

KW - C∗-algebra

KW - C0(X)-algebra

KW - multiplier algebra

KW - minimal prime ideal

KW - minimal primal ideal

KW - primitive ideal space

KW - quasi standard

U2 - 10.1007/978-3-319-18494-4_2

DO - 10.1007/978-3-319-18494-4_2

M3 - Conference contribution

SN - 978-3-319-18493-7

T3 - Operator Theory: Advances and Applications

SP - 17

EP - 29

BT - Operator Semigroups Meet Complex Analysis, Harmonic Analysis and Mathematical Physics

A2 - Arendt, Wolfgang

A2 - Chill, Ralph

A2 - Tomilov, Yuri

PB - Springer

ER -