  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, we characterize coincidence of conditional
probabilities in terms of (generalized) Doob transforms, which then leads to a
stronger classification result for the associated operator algebras in terms of
spectral radius and strong Liouville property. Furthermore, we characterize the
non-commutative peak points of the associated operator algebra in a way that
allows one to determine them from inspecting the matrix. This leads to a
concrete analogue of the maximum modulus principle for computing the norm of
operators in the ampliated operator algebras.

Operator Algebras;Probability