Decoherence for Markov chains

It is known that the subspace generated by the eigenvectors pertaining to the peripheral spectrum of any stochastic matrix is canonically equipped with a structure of a (finite-dimensional abelian) -algebra under a canonical new product introduced by E.G. Effros and M.-D. Choi. We prove that the restriction of the action of such a stochastic matrix to this subspace is indeed a -automorphism. The following new decoherence result is then established: any Markov chain encodes a conservative -dynamical system, after isolation of the persistent part from the transient one. This result gives a partial answer to the general and currently unsolved decoherence problem for a relevant class of systems.