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## Abstract

We investigate how a C*-algebra could consist of functions on a noncommutative set: a discretization of a C*-algebra A is a ∗-homomorphism A → M that factors through the canonical inclusion C(X) ⊆ ℓ∞(X) when restricted to a commutative C*-subalgebra. Any C*-algebra admits an injective but nonfunctorial discretization, as well as a possibly noninjective functorial discretization, where M is a C*-algebra. Any subhomogenous C*-algebra admits an injective functorial discretization, where M is a W*-algebra. However, any functorial discretization, where M is an AW*-algebra, must trivialize A = B(H) for any infinite-dimensional Hilbert space H.

Original language | English |
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Pages (from-to) | 19-37 |

Number of pages | 16 |

Journal | Journal of operator theory |

Volume | 77 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1 Jan 2017 |

## Keywords

- noncommutative topology
- noncommutative set
- function algebra
- discrete space
- profinite completion
- pure state
- diffuse measure
- spectrum obstruction

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## Projects

- 1 Finished