The theory of Markov chains has applications in diverse areas of research such as group theory, dynamical systems, electrical networks and information theory. These days, connections with operator algebras seem to manifest mostly in quantum information theory, where Markov chains are generalized to quantum channels.
In this talk we will show how questions about operator algebras constructed from stochastic matrices, studied by Markiewicz and myself, still motivate new problems in classical Markov chain theory. More precisely, we characterize coincidence of conditional probabilities in terms of generalized Doob transforms, which then leads to stronger classification results for the associated operator algebras. This turns out to be intimately related to determining positive harmonic functions for the stochastic matrix. Time permitting, I will explain how non-commutative peak points of the associated operator algebra can be completely characterized in terms of the stochastic matrix.
*This is based on joint work with Xinxin Chen, Langwen Hui, Christopher Linden and Yifan Zhang, conducted as part of an undergraduate research project in Illinois Geometry Lab at UIUC.