Projective unitary representations of locally compact Abelian groups as a technique for constructing covariant operator-valued measures

The M.A. Naimark theorem (1943) allows us to describe positive operator-valued measures on a measurable set $X$ as projections of projector-valued measures on $X$. To construct such measures, it is natural to consider the orbits of a projective unitary representation of some group $\mathfrak G$ acting on $X$. If such a measure is correctly defined, then it will be covariant with respect to the representation (A.S. Holevo, 1979). The report will be devoted to a specific construction in which $\mathfrak {G}$ is the direct product of a locally compact Abelian group $G$ and its dual group of characters $\hat G$. Examples will also be considered in which $\mathfrak {G}={\mathbb C}\approx {\mathbb R}\times {\mathbb R}, {\mathbb Z}_n\times {\mathbb Z}_n$ and $\mathbb {T}\times {\mathbb Z}$.