Commutative systems of covariance and a generalization of Mackey’s imprimitivity theorem.

*(English)*Zbl 0596.22003The author generalizes the Mackey imprimitivity theorem by replacing the system of imprimitivity with a system of covariance (U,a) [the reviewer, Comment. Math. Helv. 54, 629-641 (1979; Zbl 0425.22010)] which is assumed to be commutative, i.e., such that \(a(B)a(B')=a(B')a(B)\) for all B, B’ in the Borel structure considered. He shows that the group representation U is induced by a subgroup; moreover, there exists a unique probability measure which is invariant under the action of this subgroup and both uniquely determines and is determined by the system of covariance (U,a). He then applies this result in order to find a necessary and sufficient condition for the existence of an overcomplete family of coherent states [H. Scutaru, Lett. Math. Phys. 2, 101-107 (1977; Zbl 0395.22009)] for a commutative system of covariance based on a topological homogeneous space (of a second countable locally compact group) with an invariant measure.

##### MSC:

22D30 | Induced representations for locally compact groups |

28B15 | Set functions, measures and integrals with values in ordered spaces |

52A99 | General convexity |