The essentially commutative dilations of dynamical semigroups onM _n

Abstract

For identity and trace preserving one-parameter semigroups {T _t} _t≧0 on then×n-matricesM _n we obtain a complete description of their “essentially commutative” dilations, i.e., dilations, which can be constructed on a tensor product ofM _n by a commutativeW*-algebra.

We show that the existence of an essentially commutative dilation forT _t is equivalent to the existence of a convolution semigroup of probability measures ρ _t on the group Aut(M _n) of automorphisms onM _n such that$T_t = \smallint _{Aut\left( {M_n } \right)} \alpha d\rho _t \left( \alpha \right)$, and this condition is then characterised in terms of the generator ofT _t. There is a one-to-one correspondence between essentially commutative Markov dilations, weak*-continuous convolution semigroups of probability measures and certain forms of the generator ofT _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.