Absorption in invariant domains for semigroups of quantum channels
The present article concerns the notion of an absorption operator for a quantum Markov evolution, viewed as a noncommutative counterpart of classical absorption probabilities. Suppose that \(H\) is a Hilbert space and \(\Phi:B(H) \to B(H)\) is a quantum channel, i.e., a normal unital completely positive map. A projection \(p\in B(H)\) is said to be reducing for \(\Phi\) if \(\Phi(p)\geq p\); for such a \(p\) (with the space \(pH\) called also an enclosure) we define the associated absorption operator as the strong limit of the bounded increasing sequence \((\Phi^n(p))_{n=1}^\infty\); the authors include also a natural variant of this definition for a continuous evolution. It turns out that the operators \(A(p)\) encode many ergodic/recurrence/decomposition properties of the evolution. The most satisfactory results are established in the situation where \(A(p_{R^+})=1_H\), where \(R_+\) denotes the space generated by supports of all invariant normal states for the evolution; this assumption turns out to be a natural generalization of the one requiring the existence of a faithful invariant normal state. Thus, for example, if \(A(p_{R^+})=1_H\), the fixed point space of the evolution is a von Neumann algebra, coinciding with the weak\(^*\)-closed linear span of the absorption operators. Apart from developing a general theory, the authors also discuss several concrete examples, illustrating the necessity of the assumptions in the main results.
scientific article; zbMATH DE number 7376258