Doob equivalence and non-commutative peaking for Markov chains

In this paper we show how questions about operator algebras constructed from stochastic matrices motivate new results in the study of harmonic functions on Markov chains. More precisely, characterize coincidence conditional probabilities terms (generalized) Doob transforms, which then leads to a stronger classification result for associated spectral radius and strong Liouville property. Furthermore, non-commutative peak points algebra way that allows one determine them inspecting matrix. This concrete analogue maximum modulus principle computing norm operators ampliated algebras.