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

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.

28 pages, journal version of CALCO 2013 paper of the same title

References in corpus (4)

• Quantum Probability and Quantum Information Theory
• A presentation of the category of stochastic matrices
• Quantum Programs as Kleisli Maps
• Probabilistic Monads, Domains and Classical Information

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