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The essentially commutative dilations of dynamical semigroups onM n

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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.

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Communicated by R. Haag

Supported by the Deutsche Forschungsgemeinschaft

Supported by the Netherlands Organisation for the Advancement of pure research (ZWO)

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Kümmerer, B., Maassen, H. The essentially commutative dilations of dynamical semigroups onM n . Commun.Math. Phys. 109, 1–22 (1987). https://doi.org/10.1007/BF01205670

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