Classification and decomposition of quantum Markov semigroups (Q818816)

The notions of transience and recurrence play an essential role in the study of Markov processes and in probabilistic potential theory. In [\textit{F.\,Fagnola} and \textit{R.\,Rebolledo}, Probab.\ Theory Relat.\ Fields 126, No.\,~2, 289--306 (2003; Zbl 1024.60031)], transience and recurrence were defined for quantum Markov semigroups and a dichotomy was shown for irreducible semigroups. In the present paper, the transient, fast recurrent, and slow recurrent projections for quantum Markov semigroups on \(\sigma\)-finite von Neumann algebras are defined and their properties are studied. It is shown that they can be used to decompose quantum Markov semigroups as sums of ``sub''-semigroups.