  It is known that determinantal point processes have an intimate relation to
operator algebras. In the paper, we extend this relationship to encompass
dynamical aspects. Especially, we delve into two types of determinantal point
processes. One is related to discrete orthogonal polynomials of hypergeometric
type, and another is $z$-measures, which arise in the asymptotic representation
theory of the symmetric groups. Within the framework of operator algebras, we
investigate both unitary and stochastic dynamics on these point processes.
