Discretization of C*-algebras

Christiaan Heunen, Manuel L. Reyes

Research output: Contribution to journalArticlepeer-review

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 languageEnglish
Pages (from-to)19-37
Number of pages16
JournalJournal of operator theory
Volume77
Issue number1
DOIs
Publication statusPublished - 1 Jan 2017

Keywords

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

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