Existence of ground states for approximately inner two--parameter $C_0$--groups on $C^*$--algebras

In this paper‎, ‎we generalize the definitions of approximately inner $C_0$-groups and their ground states to the two‎- ‎parameter case and study necessary and sufficient conditions for a state to be ground state‎. ‎Also we prove that any approximately inner two‎- ‎parameter $C_0$--group must have at least one ground state‎. ‎Finally some applications are given‎.