  The analysis of classical consensus algorithms relies on contraction
properties of adjoints of Markov operators, with respect to Hilbert's
projective metric or to a related family of seminorms (Hopf's oscillation or
Hilbert's seminorm). We generalize these properties to abstract consensus
operators over normal cones, which include the unital completely positive maps
(Kraus operators) arising in quantum information theory. In particular, we show
that the contraction rate of such operators, with respect to the Hopf
oscillation seminorm, is given by an analogue of Dobrushin's ergodicity
coefficient. We derive from this result a characterization of the contraction
rate of a non-linear flow, with respect to Hopf's oscillation seminorm and to
Hilbert's projective metric.
