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

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. 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 also show that a commutative C^*-algebra is isomorphic to the space of convex continuous functionals from its state space to the complex numbers. This allows us to obtain an appropriately commuting state-and-effect triangle for commutative C^*-algebras.