  Let $G$ be a locally compact abelian group with a Haar measure, and $Y$ be a
measure space. Suppose that $H$ is a reproducing kernel Hilbert space of
functions on $G\times Y$, such that $H$ is naturally embedded into $L^2(G\times
Y)$ and is invariant under the translations associated with the elements of
$G$. Under some additional technical assumptions, we study the W*-algebra
$\mathcal{V}$ of translation-invariant bounded linear operators acting on $H$.
First, we decompose $\mathcal{V}$ into the direct integral of the W*-algebras
of bounded operators acting on the reproducing kernel Hilbert spaces
$\widehat{H}_\xi$, $\xi\in\widehat{G}$, generated by the Fourier transform of
the reproducing kernel. Second, we give a constructive criterion for the
commutativity of $\mathcal{V}$. Third, in the commutative case, we construct a
unitary operator that simultaneously diagonalizes all operators belonging to
$\mathcal{V}$, i.e., converts them into some multiplication operators. Our
scheme generalizes many examples previously studied by Nikolai Vasilevski and
other authors.
