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

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