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arXiv · 1708.00091

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

Arthur J. Parzygnat
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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.

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BibTeXRIS

Arthur J. Parzygnat. 2017-07-31. Discrete probabilistic and algebraic dynamics: a stochastic commutative Gelfand-Naimark Theorem. https://arxiv.org/abs/1708.00091

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