Minimal Dilations of Quantum Dynamical Semigroups to Semigroups of Endomorphisms of C*-algebras

We obtain a dilation of general one-parameter semigroups of contractive completely positive maps to semigroups of *-endomorphisms on a larger C*-algebra. The original algebra becomes a corner of this algebra. If we start with a direct sum of type-I factors this leads us to what may be called as a continuous Brattelli diagram. Here we also extend the notion of index of quantum dynamical semigroups to non-conservative case.