Abstract : This work generalize and extend results obtained recently in [2] from the Banach spaces framework to the groups framework. We study abstract classe of functions monoids for the inf-convolution struture and gives a complete description of the group of unit of such monoids. We then apply this results to obtain various versions of the Banach-Stone theorem for the inf-convolution struture in the group framework. We also give as consequence an algebraic proof of the Banach-Dieudonée theorem. Our techniques are based on a new optimization result.
