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Discrete probabilistic and algebraic dynamics: a stochastic commutative Gelfand-Naimark Theorem
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
-algebras, and the Gelfand-Naimark Theorem. No knowledge of probability
theory nor -algebras is assumed and several examples are drawn from
physics.Comment: 74 pages, 5 figures, minor edits and clarifications added since v1,
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