  We prove positivity of the Markov operators that correspond to the
hit-and-run algorithm, random scan Gibbs sampler, slice sampler and an
Metropolis algorithm with positive proposal. In all of these cases the
positivity is independent of the state space and the stationary distribution.
In particular, the results show that it is not necessary to consider the lazy
versions of these Markov chains. The proof relies on a well known lemma which
relates the positivity of the product M T M^*, for some operators M and T, to
the positivity of T. It remains to find that kind of representation of the
Markov operator with a positive operator T.
