Let G be a locally compact abelian group with Haar measure, and Y measure space. Suppose that H is reproducing kernel Hilbert space of functions on $$G\times Y$$ , such naturally embedded into $$L^2(G\times Y)$$ invariant under the translations associated elements G. Under some additional technical assumptions, we study W*-algebra $${\mathcal {V}}$$ translation-invariant bounded linear operator...