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## Abstract

A C*-algebra is determined to a great extent by the partial order of its commutative C*-subalgebras. We study order-theoretic properties of this dcpo. Many properties coincide: the dcpo is, equivalently, algebraic, continuous, meet-continuous, atomistic, quasi-algebraic, or quasi-continuous, if and only if the C*-algebra is scattered. For C*-algebras with enough projections, these properties are equivalent to finite-dimensionality. Approximately finite-dimensional elements of the dcpo correspond to Boolean subalgebras of the projections of the C*-algebra. Scattered C*-algebras are finite-dimensional if and only if their dcpo is Lawson-scattered. General C*-algebras are finite-dimensional if and only if their dcpo is

order-scattered.

order-scattered.

Original language | English |
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Pages (from-to) | 972-1006 |

Number of pages | 41 |

Journal | Mathematical Structures in Computer Science |

Volume | 29 |

Issue number | 7 |

Early online date | 21 Mar 2019 |

DOIs | |

Publication status | Published - Aug 2019 |

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## Projects

- 1 Finished