On the range of the generator of a quantum Markov semigroup (Q2790332)

It is known that every bounded self-adjoint derivation of \(B(h)\), where \(h\) is a Hilbert space, is inner, that is of the form \(\delta(x) = i(Hx - xH)\), \(x \in B(h)\), where \(H\) is a bounded self-adjoint operator. This can be used to show that the range of a generator of a norm continuous automorphism group on \(B(h)\) has a trivial commutant.NEWLINENEWLINEIn the paper under review, the authors discuss a similar problem for a generator of a norm continuous semigroup of unital completely positive normal maps on \(B(h)\) preserving a fixed normal state. The main result says that if \(L\) is the generator of such a (nontrivial) semigroup, and the fixed point subalgebra of the semigroup is atomic, then the commutant of the range of \(L\) is trivial. This has interesting consequences for different versions of the irreversible \((H, \beta)\)-KMS condition, introduced in the article by \textit{L. Accardi}, the second author and \textit{R. Quezada} [`` Weighted detailed balance and local KMS condition for non-equilibrium stationary states'', Bussei Kenkyu 97, 318--356 (2011)]. The authors present also certain non-trivial finite-dimensional examples.