In this chapter, we define and explore basic properties of a quantum Markov semigroup {It, t ≥ 0} of linear maps on the C* -algebra or von Neumann algebra A that characterizes the quantum system. The quantum Markov semigroup (QMS) plays a key role in describing quantum Markov processes that are to be explored in the subsequent chapters. The concept of QMS extends the semigroup of probability transition operators {Tt, t ≥ 0} for a classical Markov process to its noncommutative counterpart. Suppose the classical Markov process {Xt, t ≥ 0} is defined on the complete filtered classical probability space (Ω, Ƒ, P; {Ƒt, t ≥ 0}) and with values in a measurable space (핏, B (X)). Recall that for each t ≥ 0, the probability transition operator Tt: L∞ (X, B (X)) → L∞ (X, B (X)) by

(Ttf)(Xs) = Ex[f(Xs+t) | Ƒs] ∀s, t ≥ 0.

A semigroup of linear maps on the C* -algebra or von Neumann algebra A is said to be a quantum dynamical semigroup (QDS) if (i) I0 = J (the identity operator on A); (ii) ItIs = It+s for all t, s ≥ 0; (iii) It is completely positive for each t ≥ 0; and (iv) It is σ weakly continuous on A, i.e., a ↦ tr(ρIt (a)) is continuous from A to ℂ for each ρ ∈ S (A) (the space of quantum states) and for each t ≥ 0. If in addition It (I) = I, (respectively, It (I) ≤ I) for all t ≥ 0, then the QDS is said to be a quantum Makrov semigroup (QMS) (respectively, quantum sub-Markov semigroup).