From Kleisli Categories to Commutative C*-algebras: Probabilistic Gelfand Duality

Robert Furber, Bart Jacobs

Research output: Contribution to journalArticlepeer-review

Abstract

C*-algebras form rather general and rich mathematical structures that can be studied with different morphisms (preserving multiplication, or not), and with different properties (commutative, or not). These various options can be used to incorporate various styles of computation (set-theoretic, probabilistic, quantum) inside categories of C*-algebras. At first, this paper concentrates on the commutative case and shows that there are functors from several Kleisli categories, of monads that are relevant to model probabilistic computations, to categories of C*-algebras. This yields a new probabilistic version of Gelfand duality, involving the "Radon" monad on the category of compact Hausdorff spaces. We then show that the state space functor from C*-algebras to Eilenberg-Moore algebras of the Radon monad is full and faithful. This allows us to obtain an appropriately commuting state-and-effect triangle for C*-algebras.
Original languageEnglish
Pages (from-to)1-28
Number of pages28
JournalLogical Methods in Computer Science
Volume11
Issue number2
DOIs
Publication statusPublished - 10 Jun 2015

Keywords

  • Mathematics - Category Theory
  • Computer Science - Logic in Computer Science

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