Intertwining Properties for Bimodule Quantum Markov Semigroups

In this paper, we study the Bakry-Émery estimates for GNS- and KMS-symmetric semigroups in terms of the Fourier multiplier of the gradient form and the iterated gradient form in the framework of quantum Fourier analysis. We also systematically investigate the intertwining properties for bimodule GNS- and KMS-symmetric quantum Markov semigroups and compare with the Bakry-Émery estimates. A number of examples of GNS- and KMS-symmetric semigroups satisfying these intertwining properties are presented.