Abstract
We introduce quantum Markov states (QMS) in a general tree graph G=(V,E), extending the Cayley tree's case. We investigate the Markov property w.r.t. the finer structure of the considered tree. The main result of the present paper concerns the diagonalizability of a locally faithful QMS phi on a UHF-algebra AV over the considered tree by means of a suitable conditional expectation into a maximal abelian subalgebra. Namely, we prove the existence of a Umegaki conditional expectation E:AV -> DV such that phi=phi remvoeDV degrees E. Moreover, it is clarified the Markovian structure of the associated classical measure on the spectrum of the diagonal algebra DV.