A transition operator T is a
nonnegative contraction on an AL space L such that ∥Tμ∥ = ∥μ∥ for μ ≧ 0. The set
ℳ = {f ∈ L∗: T∗f = f} of invariant functions of the adojoint T∗ turns out to be
lattice isomorphic to C(B) for a certain hyperstonian compact Hausdorff B. For
the transition operator of a countable state Markov chain, B is the Feller
boundary of the process, and in the general case we call B the Feller boundary
induced by T. For the general case we exhibit several Markov processes
associated with T such that B appears as a subset of the state space. These
processes involve the potential theory of T∗. When L is separable there is a
quotient space B0 of B and a measure μ0 with B0 as closed support such
that ℳ is isomorphic to L∞(B0,μ0). There is also a Markov process whose
paths converge to B0 with probability 1. However, we do not obtain the
kernel representation of superharmonic functions as with the Martin-Doob
boundary.
