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

2017/07/31 by Parzygnat, Arthur J. · 1 citation
#18A40 (Secondary) #60B05 (Primary) 47B65 #Category Theory (math.CT) #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Physics (math-ph) #Probability (math.PR)

paper · doi:10.48550/arxiv.1708.00091

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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