K-property for Maharam extensions of nonsingular Bernoulli and Markov shifts

It is shown that each conservative nonsingular Bernoulli shift is either of type $I{I}_1$ or $II{I}_1$. Moreover, in the latter case the corresponding Maharam extension of the shift is a $K$-automorphism. This extends earlier results obtained by Z.~Kosloff for the equilibrial shifts. Nonequilibrial shifts of type $II{I}_1$ are constructed. We further generalize (partly) the main results to nonsingular Markov shifts.