For identity and trace preserving one-parameter semigroups (T sub t) t>0 on the nxn-matrices M sub n, the authors obtain a complete description of their 'essentially commutative' dilations, i.e., dilations, which can be constructed on a tensor product of Mn by a commutative W star-algebra. The authors show that the existence of an essentially commutative dilation for T sub t is equivalent to the existence of a convolution semigroup of probability measures the sub t on the group Aut(Mn) of automorphisms on Mn, such that and this condition is then characterized in terms of the generator of T sub t. There is a one-to-one correspondence between essentially commutative Markov dilations, weak star -continuous convolution semigroups of probability measures, and certain forms of the generator of T sub t. In particular, certain dynamical semigroups which do not satisfy the detailed balance condition are shown to admit a dilation. This provides the first example of a dilation for such a semigroup.
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