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

TL;DR

This paper develops a probabilistic extension of Gelfand duality for commutative C*-algebras using Kleisli categories and the Radon monad, revealing new categorical structures for probabilistic computation.

Contribution

It introduces a new probabilistic Gelfand duality linking Kleisli categories of relevant monads to C*-algebras, with a full and faithful state space functor.

Findings

Establishes a functor from Kleisli categories to C*-algebras.

Proves the state space functor is full and faithful.

Derives a commuting state-and-effect triangle for C*-algebras.

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.