On $C^*$-algebras generated by the set of probability distributions

In the paper we consider the $C^*$-algebras generated by the Markov operators, which are induced by the probabilistic dynamical system. It is shown that in some cases these algebras are isomorphic to the Toeplitz, Kunz, and Kunz-Toeplitz algebras. By the probabilistic dynamic system, the Markov operator is constructed, and it is shown that it can be extended to a quantum channel on a convex set of positive operators with the unit trace.