Given a family of transition probability functions between measure spaces and an initial distribution Kolmogorov’s existence theorem associates a unique Markov process on the product space. Here a canonical non-commutative analogue of this result is established for families of completely positive maps betweenC* algebras satisfying the Chapman-Kolmogorov equations. This could be the starting point for a theory of quantum Markov processes.

Given a contractive tuple of Hilbert space operators satisfying certainA-relations we show that there exists a unique minimal dilation to generators of Cuntz-Krieger algebras or its extension by compact operators. This Cuntz-Krieger dilation can be obtained from the classical minimal isometric dilation as a certain maximalA-relation piece. We define a maximal piece more generally for a finite set of polynomials inn noncommuting variables. We classify all representations of Cuntz-Krieger algebrasO_A obtained from dilations of commuting tuples satisfyingA-relations. The universal properties of the minimal Cuntz-Krieger dilation and the WOT-closed algebra generated by it is studied in terms of invariant subspaces.