doi:10.1088/1751-8121/aceed5engBouthelier-Madre, C.Clemente-Gallardo, J.González-Bravo, L.Martínez-Crespo, D.Hybrid Koopman $C^*$–formalism and the hybrid quantum–classical master equation*ART-2023-135197Based on Koopman formalism for classical statistical mechanics, we propose a formalism to define hybrid quantum–classical dynamical systems by defining (outer) automorphisms of the $C^*$ algebra of hybrid operators and realizing them as linear transformations on the space of hybrid states. These hybrid states are represented as density matrices on the Hilbert space obtained from the hybrid $C^*$–algebra by the GNS construction. We also classify all possible dynamical systems which are unitary and obtain the possible hybrid Hamiltonian operators.2023http://zaguan.unizar.es/record/12803810.1088/1751-8121/aceed5http://zaguan.unizar.es/record/128038oai:zaguan.unizar.es:128038info:eu-repo/grantAgreement/ES/AEI/PID2021-123251NB-I00info:eu-repo/grantAgreement/ES/DGA-CUS/581-2020info:eu-repo/grantAgreement/ES/DGA/E48-23Rinfo:eu-repo/grantAgreement/ES/DGA-IIU/1408-2018Journal of Physics A-Mathematical and Theoretical 56, 37 (2023), 374001 [21 pp.]All rights reservedhttp://www.europeana.eu/rights/rr-f/info:eu-repo/semantics/openAccess