The adjoints of dilation operators on H-cones

If S is an H-cone and P:S -->S is a localizable dilation operator on S (i.e ., P is additive increasing, contractive, continuous in order from below an d s boolean AND (Ps+t-Pt+Pf)is an element ofS, s,t is an element ofS,f is an element of (S-S)(+)), then it is proved that its adjoint P*:S*-->S* (i.e., P*mu=mu degreesP) is a lso a localizable dilation operator. This is an improvement of a result obt ained by G. Mokobodzki in the frame of excessive functions.