A remark on invariance of quantum Markov semigroups

In [3, 9], using the theory of noncommutative Dirichlet forms in the sense of Cipriani [6] and the symmetric embedding map, authors constructed the KMS-symmetric Markovian semigroup {S_t}_t≥o on a von Neumann algebra M with an admissible function f and an operator x ∈ M. We give a sufficient and necessary condition for x so that the semigroup {S_t}_t≥o acts separately on diagonal and off-diagonal operators with respect to a basis and study some results.