Discrete probabilistic and algebraic dynamics: a stochastic commutative Gelfand-Naimark Theorem

TL;DR

This paper develops a stochastic analogue of the Gelfand-Naimark Theorem, connecting algebraic, topological, and probabilistic concepts through a new category of stochastic maps on compact spaces.

Contribution

It introduces a category of stochastic maps, constructs a stochastic Gelfand spectrum functor, and proves a stochastic Gelfand-Naimark Theorem, bridging algebra, topology, and probability.

Findings

Established a stochastic version of the Gelfand-Naimark Theorem

Constructed a stochastic spectrum functor for compact Hausdorff spaces

Connected concepts from algebra, operator theory, topology, and probability

Abstract

We introduce a category of stochastic maps (certain Markov kernels) on compact Hausdorff spaces, construct a stochastic analogue of the Gelfand spectrum functor, and prove a stochastic version of the commutative Gelfand-Naimark Theorem. This relates concepts from algebra and operator theory to concepts from topology and probability theory. For completeness, we review stochastic matrices, their relationship to positive maps on commutative $C^*$-algebras, and the Gelfand-Naimark Theorem. No knowledge of probability theory nor $C^*$-algebras is assumed and several examples are drawn from physics.